콕시터 표기법

반사 3D 포인트 그룹의 기본 영역

, [ ]=[1] C1v	, [2] C2v	, [3] C3v	, [4] C4v	, [5] C5v	, [6] C6v
주문2길	주문4길	순서 6	주문 8	주문 10	순서 12
[2]=[2,1] D1h	[2,2] D2h	[2,3] D3h	[2,4] D4h	[2,5] D5h	[2,6] D6h
주문4길	주문 8	순서 12	순서 16	주문 20	주문24
, [3,3], Td		, [4,3], Oh		, [5h,3] I
주문24		주문로48번길		주문로120번길
Coxeter 표기법은 다면체 그룹과 같이 Coxeter 다이어그램의 분기 순서 목록으로 Coxeter 그룹을 표현한다. = [p,q]. 다이헤드 그룹 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 명시적인 순서 2 가지 , [2,n] 가지와

기하학에서 Coxeter 표기법(Coxeter 기호도 역시 Coxeter 기호)은 대칭군 분류 체계로서 Coxeter-Dynkin 도표의 구조를 나타내는 브라켓형 표기법으로 Coxeter 그룹의 근본적인 반사 사이의 각도를 기술하고 있으며, 특정 부분군을 나타내는 수식어를 사용한다. 이 표기법은 H. S. M. Coxeter의 이름을 따서 명명되었으며, 노먼 존슨에 의해 보다 포괄적으로 정의되었다.

반사 그룹

순수한 반사로 정의되는 Coxeter 그룹의 경우, 대괄호 표기법과 Coxeter-Dynkin 도표 사이에 직접적인 일치성이 있다. 괄호 표기법의 숫자는 Coxeter 도표의 분기에 있는 거울 반사 순서를 나타낸다. 직교 거울 사이에 2초를 억제하면서 동일한 단순화를 사용한다.

Coxeter 표기법은 선형 다이어그램에 대해 행에 있는 가지 수를 나타내기 위해 지수를 사용하여 단순화된다. 따라서 A 그룹은_n [3ⁿ⁻¹]로 표시되며, 이는 n-1 주문-3 분기로 연결된 n개의 노드를 의미한다. 예제 A₂ = [3,3] = [3²] 또는 [3^1,1]은 도표 또는 를 나타낸다.

Coxeter는 처음에는 숫자의 수직 위치를 갖는 분기 도표를 나타냈으나, 나중에 [...,3^p,q] 또는 [3^p,q,r]과 같은 지수 표기법으로 약칭하여 [3^1,1,1] 또는 [3^1,1] = 또는 D로₄ 시작한다. A_n 계열에 맞는 특별한 경우로서 0에 대해 허용되는 콕시터(예: A₃ = [3,3,3] = [3^4,0,04,0] = [3^3,1] = [3] = [3^2,2], like = = = .

주기 다이어그램에 의해 형성된 콕시터 그룹은 삼각형 그룹(pq r)에 대한 [(p,q,r)] =와 같이 괄호 안에 있는 괄호로 표현된다. 분기 순서가 같을 경우, [(3,3,3^[4],3)] = [3]와 같이 괄호 안의 주기 길이에 따라 지수로 그룹화할 수 있다. Coxeter 다이어그램 또는 .를 나타내는 것은 [3,3,3)] 또는 [3,3^[3]]로 나타낼 수 있다.

더 복잡한 루프 도표도 주의 깊게 표현할 수 있다. 파라콤팩트 콕시터 그룹은 콕시터 표기법[(3,3,3)]으로 나타낼 수 있으며, 두 개의 인접한 [(3,3,3) 루프를 보여주는 내포/과대칭 괄호를 가지고 있으며, 콕시터 다이어그램의 롬빅 대칭을 나타내는 [3^{[ ]×[ ]}]으로 보다 압축적으로 표현된다. 파라콤팩트 전체 그래프 다이어그램 또는 는 일반 사면체 코엑스터 다이어그램의 대칭으로 위첨자 [3,3]와 함께 [3^[3,3]]로 표시된다.

Coxeter 다이어그램은 보통 순서-2 가지를 사용하지 않고 그대로 두지만, 괄호 표기법에는 서브그래프를 연결하기 위한 명시적 2가 포함되어 있다. 따라서 Coxeter 다이어그램 = A₂×A₂ = 2A는₂ [3]×[3] = [3]² = [3,2,3]로 나타낼 수 있다. 때때로 명시적 2-브랜치는 2개의 라벨 또는 간격이 있는 선과 함께 포함되거나 [3,2,3]과 동일한 표시로 포함될 수 있다.

유한군 순위 그룹 심볼 브래킷 표기법 콕시터 도표를 만들다 2 A2 [3] 2 B2 [4] 2 H2 [5] 2 G2 [6] 2 아이2(p) [p] 3 나h, H3 [5,3] 3 Td, A3 [3,3] 3 Oh, B3 [4,3] 4 A4 [3,3,3] 4 B4 [4,3,3] 4 D4 [31,1,1] 4 F4 [3,4,3] 4 H4 [5,3,3] n An [3n−1] .. n Bn [4,3n−2] ... n Dn [3n−3,1,1] ... 6 E6 [32,2,1] 7 E7 [33,2,1] 8 E8 [34,2,1]

아핀 그룹 그룹 심볼 브래킷 표기법 콕시터 다이어그램 ${\displaystyle {\tilde{I}_{1},{\tilde{A}_{1}}$ [∞] ${\displaystyle {\tilde{A}_{2}}:$ [3[3]] ${\displaystyle {\tilde{C}_{2}}:$ [4,4] ${\displaystyle {\tilde{G}_{2}}$ [6,3] ${\displaystyle {\tilde{A}_{3}}$ [3[4]] ${\displaystyle {\tilde{B}_{3}}$ [4,31,1] ${\displaystyle {\tilde{C}_{3}}$ [4,3,4] ${\displaystyle {\tilde{A}_{4}}$ [3[5]] ${\displaystyle {\tilde{B}_{4}}$ [4,3,31,1] ${\displaystyle {\tilde{C}_{4}}$ [4,3,3,4] ${\displaystyle {\tilde{D}_{4}}$ [ 31,1,1,1] ${\displaystyle {\tilde{F}_{4}}$ [3,4,3,3] ${\displaystyle {\tilde{A}_{n}}$ [3[n+1]] ... 또는 ... ${\displaystyle {\tilde{B}_{n}}$ [4,3n−3,31,1] ... ${\displaystyle {\tilde{C}_{n}}$ [4,3n−2,4] ... ${\displaystyle {\tilde{D}_{n}}$ [ 31,1,3n−4,31,1] ... ${\displaystyle {\tilde{E}_{6}}$ [32,2,2] ${\displaystyle {\tilde{E}_{7}}$ [33,3,1] ${\displaystyle {\tilde{E}_{8}=E_{9}}$ [35,2,1]

쌍곡선군 그룹 심볼 브래킷 표기법 콕시터 도표를 만들다 [p,q] 2(p + q) < pq와 함께 [(p,q,r)] + + < 1 ${\displaystyle {\frac {1}{p}+{$p}+{$p}{$p}{p$}}+{\frac${1$}{r}<1}}$ $디스플레이 스타일 {\overline {BH}}_{3}}}$ [4,3,5] ${\displaystyle {\overline{K}_{3}}$ [5,3,5] ${\displaystyle {\overline{J}_{3},{\tilde{H}_{3}}}$ [3,5,3] ${\displaystyle {\overline{DH}_{3}}$ [5,31,1] ${\displaystyle {\widehat{AB}_{3}}$ [(3,3,3,4)] ${\displaystyle {\widehat{AH}_{3}}$ [(3,3,3,5)] ${\displaystyle {\widehat{BB}_{3}}$ [(3,4,3,4)] ${\displaystyle}\widehat {BH}}_{3}}}$ [(3,4,3,5)] $디스플레이 스타일 {\widehat {\displaystyle}HH}}_{3}}}$ [(3,5,3,5)] ${\displaystyle {\overline {H}_{4},{\tilde {H}_{4},H_{5}}}$ [3,3,3,5] $디스플레이 스타일 {\overline {BH}}_{4}}}$ [4,3,3,5] ${\displaystyle {\overline{K}_{4}}$ [5,3,3,5] ${\displaystyle {\overline{DH}_{4}}$ [5,3,31,1] ${\displaystyle {\widehat{AF}_{4}}$ [(3,3,3,3,4)]

어핀 및 쌍곡선 그룹의 경우, 첨자는 각 그룹이 유한 그룹의 다이어그램에 노드를 추가하여 얻었기 때문에 각각의 경우에서 노드 수보다 1개 적다.

부분군

Coxeter의 표기법은 대괄호 밖에 상위첨자 연산자를 추가하여 회전/변환 대칭을 나타내며, 이 연산자는 그룹 [X]⁺의 순서를 반으로 자르고, 따라서 지수 2 하위그룹을 나타낸다. 이 연산자는 반사를 회전(또는 번역)으로 대체하여 짝수 연산자를 적용해야 함을 의미한다. Coxeter 그룹에 적용했을 때, 이것을 다이렉트 서브그룹이라고 부르는데, 남아 있는 것은 반사 대칭이 없는 다이렉트 이소메트리일 뿐이다.

연산자는 또한 [X,Y⁺] 또는 [X,(Y,Z)]⁺와 같이 괄호 안에 적용될 수 있으며 반사 발전기와 비반사 발전기를 모두 포함할 수 있는 "semidirect" 하위그룹을 생성할 수 있다. 반간접적인 하위 그룹은 그것과 인접한 주문 분기가 있는 Coxeter 그룹 하위 그룹에만 적용할 수 있다. Coxeter 그룹 내부의 괄호들에 의한 요소는 인접한 순서의 가지를 반순으로 나누는 효과를 가지면서 위첨자 연산자에게 줄 수 있으므로, 일반적으로 짝수 숫자로만 적용된다. 예를 들면 [4,3⁺]과 [4,(3,3)]()⁺이다.

인접한 홀수 분기를 적용하면 지수 2의 하위 그룹을 생성하지 않고, 대신 [5,1⁺] = [5/2]와 같이 겹치는 기본 도메인을 생성하여 펜타그램, {5/2}, [5,3⁺]와 같은 이중으로 포장된 폴리곤을 슈바르츠 삼각형 [5/2,3] 밀도 2와 관련시킬 수 있다.

순위 2 그룹의 예

그룹		주문	제너레이터	부분군		주문	제너레이터	메모들
[p]		2p	{0,1}	[p]+		p	{01}	직접 부분군
[2p+] = [2p]+		2p	{01}	[2p+]+ = [2p]+2 = [p]+		p	{0101}
[2p]		4p	{0,1}	[1+,2p] = [p]	= =	2p	{101,1}	부분군 반
				[2p,1+] = [p]	= =		{0,010}
				[1+,2p,1+] = [2p]+2 = [p]+	= =	p	{0101}	쿼터 그룹

주변 요소가 없는 그룹은 균일한 폴리토프의 링 노드인 Coxeter-Dynkin 도표에서 볼 수 있으며 벌집합은 요소 주위의 구멍 노드, 대체 노드가 제거된 빈 원과 관련이 있다. 그래서 스너브 큐브는 대칭 [4,3]()⁺을 가지며 스너브 4면체는 대칭 [4,3⁺]()을 가지며, 데미큐브, h{4,3} = {3,3}( 또는 = )은 대칭 [1⁺,4,3] = [3,3]( 또는 = )이다.

참고: Pyritohedral 대칭은 명확성을 위해 그래프를 공백으로 분리하고, Coxeter 그룹에서 {0,1,2} 발전기를 사용하여 Pyritohedral 생성기 {0,12}, 반사 및 3배 회전으로 기록할 수 있다. 그리고 치랄 사면 대칭은 또는 , [1⁺,4,3⁺] = [3,3]⁺로 표기할 수 있으며 발전기는 {12,0120}이다.

부분군 및 확장 그룹 절반 분할

반감 작업 예제

[1,4,1] = [4]	= = [1+,4,1]=[2]=[ ]×[ ]
= = [1,4,1+]=[2]=[ ]×[ ]	= = = [1+,4,1+] = [2]+

존슨은 거울을 제거하는 자리 표시자⁺ 1 노드로 작업하도록 연산자를 확장하여 기본 도메인의 크기를 두 배로 늘리고 그룹 순서를 반으로 줄인다.^[1] 일반적으로 이 작업은 짝수 분기로 경계된 개별 미러에만 적용된다. 1은 거울을 나타내므로 [2p]을 도표나 와 같이 [2p,1], [1,2p] 또는 [1,2p,1]로 볼 수 있다. 미러 제거의 효과는 Coxeter 다이어그램: = , 또는 괄호 표기법:[1⁺,2p, 1] = [1,p,1] = [p]에서 볼 수 있는 연결 노드를 복제하는 것이다.

이러한 거울은 각각 h[2p] = [1⁺,2p,1] = [1,2p,1⁺] = [p], 반사 부분군 지수 2가 되도록 제거할 수 있다. 이는 노드 위에 기호를 추가하여 Coxeter 다이어그램에 표시할 수 있다: = = .

두 미러가 모두 제거되면 분기 순서가 순서의 절반인 회전 지점이 되는 1/4 부분군이 생성된다.

q[2p] = [1⁺,2p,1⁺] = [p],⁺ 지수 4. = = = = = = .

예를 들어 (p=2 포함): [4,1⁺] = [1⁺,4] = [2] = [ ]×[], 순서 4. [1⁺,4,1⁺] = [2],⁺ 순서 2.

반감과는 반대로 거울을 추가하는 두 배^[2], 기본 영역을 이등분하고, 그룹 순서를 두 배로 하는 것이다.

[[p] = [2p]

사면 대칭은 팔면 그룹의 절반 그룹인 h[4,3] = [1⁺,4,3] = [1,4,3] = [3,3]과 같이 반감 연산이 상위 랭크 그룹에 적용된다. 미러 제거의 효과는 Coxeter 다이어그램에서 볼 수 있는 모든 연결 노드를 복제하는 것이다. = , h[2p,3] = [1⁺,2p,3] = [(p,3,3)]

노드가 인덱싱된 경우, 절반의 부분군에는 합성물로 새 미러를 표시할 수 있다. 과 마찬가지로 발전기 {0,1}에는 부분군 = , 생성기 {1,010}이(가) 있으며, 여기서 미러 0은 제거되고 미러 0에 반사된 미러 1의 복사본으로 대체된다. 또한 , 생성기 {0,1,2}을(를) 지정하면 절반 그룹 = , 생성기 {1,2,010}이(가) 있음.

거울을 추가하여 반감하는 동작도 역방향으로 적용한다: [[3,3] = [4,3] = [4,3], 또는 더 일반적으로 [[(q,q,p)] = [2p,q]

사면 대칭	팔면 대칭
Td, [3,3] = [1+,4,3] = = (주문 24)	Oh, [4,3] = [3,3] (주문 48)

급진적 부분군

존슨은 또한 운영자와 유사하게 작용하지만 회전 대칭은 제거하는 "^[3]라디칼" 부분군을 위해 별표나 별 * 연산자를 추가했다. 급진적인 부분군의 지수는 제거된 원소의 순서다. 예를 들면 [4,3*] ≅ [2,2]이다. 제거된 [3] 부분군은 순서 6이므로 [2,2]는 [4,3]의 지수 6 부분군이다.

급진적인 부분군은 확장된 대칭 연산에 대한 역연산을 나타낸다. 예를 들어 [4,3*] ≅ [2,2], 역[2,2] as [4,3]으로 확장할 수 있다. 부분군은 Coxeter 다이어그램 또는 ≅으로 표현할 수 있다. 제거된 노드(미러)는 인접한 미러 가상 미러를 실제 미러로 만든다.

If [4,3] has generators {0,1,2}, [4,3⁺], index 2, has generators {0,12}; [1⁺,4,3] ≅ [3,3], index 2 has generators {010,1,2}; while radical subgroup [4,3*] ≅ [2,2], index 6, has generators {01210, 2, (012)³}; and finally [1⁺,4,3*], index 12 has generators {0(12)²0, (012)²01}.

삼온 부분군

삼온 부분군은 지수 3 부분군이다. 존슨씨는 연산자 ⅄, 지수 3으로 트리오닉 하위그룹을 정의하고 있다. 순위 2 Coxeter 그룹의 경우 [3], 트리오닉 하위 그룹의 경우 [3^⅄]은 [ ], 단일 미러다. 그리고 [3p]의 경우, 삼온 부분군은 [3p]^⅄ ≅ [p]이다. 생성자가 {0,1}인 경우 에는 3개의 트리온 하위 그룹이 있다. ⅄ 기호를 제거할 미러 발생기 옆에 놓거나, 두 가지 모두에 대해 분기([3p,1^⅄] = , ,, [3p^⅄] = {0,101}, {0101,1} 또는 {101,010})로 구분할 수 있다.

사면 대칭의 삼면체 부분군: [3,3]^⅄ ≅ [2⁺,4], 정규 사면체 및 사면체 디스페노이드의 대칭과 관련된다.

3위 Coxeter 그룹 [p,3]의 경우, [p,3^⅄] or [p/2,p] 또는 =. 예를 들어 유한 그룹 [4,3^⅄] ≅ [2,4], 유클리드 그룹 [6,3^⅄] ≅ [3,6], 쌍곡 그룹 [8,8^⅄]이 있다.

홀수 순서 인접 분기인 p는 그룹 순서를 낮추지 않고 중복되는 기본 도메인을 만든다. 그룹 순서는 그대로인 반면 밀도는 높아진다. 예를 들어, 정규 다면체의 고면체 대칭인 [5,3]은 2개의 일반 다면체의 대칭인 [5/2,5]가 된다. 또한 쌍곡선 기울기 {p,3} 및 별 쌍곡선 기울기 {p/2,p}과(와) 관련이 있다.

순위 4, [q,2p,3^⅄] = [2p,(p,q,q)], = .

예를 들어, [3,4,3^⅄]의 [3,3,3] 또는 [3,3]의 발전기 {0,1,3,3] 생성기 {0,1,2,32123}이(가) 있는 [3,4,3]의 생성기 {0,1,2,3}을(를) 예로 들 수 있다. 쌍곡선 그룹의 경우 [3,6,3^⅄] = [6,3^[3]], [4,4^⅄,3] = [4,4,4].

사면 대칭의 삼면체 부분군

Johnson은 [3,3]의 두 개의 특정 삼온 하위그룹을^[4] 식별했고, 먼저 지수 3 하위그룹 [3,3]^⅄ ≅ [2⁺,4], [3,3] ( = = ) 생성기가 {0,1,2}인 것을 확인했다. 또한 발전기 {02,1}을(를) 상기시키기 위해 [(3,3^⅄,2)]()으로 쓸 수 있다. 이 대칭 감소는 정규 4면체와 4각형 디스페노이드 사이의 관계로서, 두 개의 반대쪽 가장자리에 수직인 4면체의 스트레칭을 나타낸다.

둘째로, 그는 관련 지수 6 하위그룹[3,3] 또는 [(3,3,^Δ2^⅄)]⁺ (), [3,3]⁺ ≅[2,2]⁺의 지수 3을 확인하고, [3,3]의 발전기 {02,1021}, 그리고 그 발전기 {01,1,2}을(를) 식별한다.

또한 이러한 부분군은 인접한 분기가 모두 짝수인 [3,3] 부분군이 있는 더 큰 Coxeter 그룹 내에서 적용된다.

For example, [(3,3)⁺,4], [(3,3)^⅄,4], and [(3,3)^Δ,4] are subgroups of [3,3,4], index 2, 3 and 6 respectively. The generators of [(3,3)^⅄,4] ≅ [[4,2,4]] ≅ [8,2⁺,8], order 128, are {02,1,3} from [3,3,4] generators {0,1,2,3}. And [(3,3)^Δ,4] ≅ [[4,2⁺,4]], order 64, has generators {02,1021,3}. As well, [3^⅄,4,3^⅄] ≅ [(3,3)^⅄,4].

Also related [3^1,1,1] = [3,3,4,1⁺] has trionic subgroups: [3^1,1,1]^⅄ = [(3,3)^⅄,4,1⁺], order 64, and 1=[3^1,1,1]^Δ = [(3,3)^Δ,4,1⁺] ≅ [[4,2⁺,4]]⁺, order 32.

Central inversion

A central inversion, order 2, is operationally differently by dimension. The group [ ]ⁿ = [2ⁿ⁻¹] represents n orthogonal mirrors in n-dimensional space, or an n-flat subspace of a higher dimensional space. The mirrors of the group [2ⁿ⁻¹] are numbered ${\displaystyle 0\dots n-1}$. The order of the mirrors doesn't matter in the case of an inversion. The matrix of a central inversion is ${\displaystyle -I}$, the Identity matrix with negative one on the diagonal.

From that basis, the central inversion has a generator as the product of all the orthogonal mirrors. In Coxeter notation this inversion group is expressed by adding an alternation ⁺ to each 2 branch. The alternation symmetry is marked on Coxeter diagram nodes as open nodes.

A Coxeter-Dynkin diagram can be marked up with explicit 2 branches defining a linear sequence of mirrors, open-nodes, and shared double-open nodes to show the chaining of the reflection generators.

For example, [2⁺,2] and [2,2⁺] are subgroups index 2 of [2,2], , and are represented as (or ) and (or ) with generators {01,2} and {0,12} respectively. Their common subgroup index 4 is [2⁺,2⁺], and is represented by (or ), with the double-open marking a shared node in the two alternations, and a single rotoreflection generator {012}.

Dimension	Coxeter notation	Order	Coxeter diagram	Operation	Generator
2	[2]+	2		180° rotation, C2	{01}
3	[2+,2+]	2		rotoreflection, Ci or S2	{012}
4	[2+,2+,2+]	2		double rotation	{0123}
5	[2+,2+,2+,2+]	2		double rotary reflection	{01234}
6	[2+,2+,2+,2+,2+]	2		triple rotation	{012345}
7	[2+,2+,2+,2+,2+,2+]	2		triple rotary reflection	{0123456}

Rotations and rotary reflections

Rotations and rotary reflections are constructed by a single single-generator product of all the reflections of a prismatic group, [2p]×[2q]×... where gcd(p,q,...)=1, they are isomorphic to the abstract cyclic group Z_n, of order n=2pq.

The 4-dimensional double rotations, [2p⁺,2⁺,2q⁺] (with gcd(p,q)=1), which include a central group, and are expressed by Conway as ±[C_p×C_q],^[5] order 2pq. From Coxeter diagram , generators {0,1,2,3}, requires two generator for [2p⁺,2⁺,2q⁺], as {0123,0132}. Half groups, [2p⁺,2⁺,2q⁺]⁺, or cyclic graph, [(2p⁺,2⁺,2q⁺,2⁺)], expressed by Conway is [C_p×C_q], order pq, with one generator, like {0123}.

If there is a common factor f, the double rotation can be written as 1⁄f[2pf⁺,2⁺,2qf⁺] (with gcd(p,q)=1), generators {0123,0132}, order 2pqf. For example, p=q=1, f=2, 1⁄2[4⁺,2⁺,4⁺] is order 4. And 1⁄f[2pf⁺,2⁺,2qf⁺]⁺, generator {0123}, is order pqf. For example, 1⁄2[4⁺,2⁺,4⁺]⁺ is order 2, a central inversion.

In general a n-rotation group, [2p₁⁺,2,2p₂⁺,2,...,p_n⁺] may require up to n generators if gcd(p₁,..,p_n)>1, as a product of all mirrors, and then swapping sequential pairs. The half group, [2p₁⁺,2,2p₂⁺,2,...,p_n⁺]⁺ has generators squared. n-rotary reflections are similar.

Examples

Dimension	Coxeter notation	Order	Coxeter diagram	Operation	Generators	Direct subgroup
2	[2p]+	2p		Rotation	{01}	[2p]+2 = [p]+	Simple rotation: [2p]+2 = [p]+ order p
3	[2p+,2+]			rotary reflection	{012}	[2p+,2+]+ = [p]+
4	[2p+,2+,2+]			double rotation	{0123}	[2p+,2+,2+]+ = [p]+
5	[2p+,2+,2+,2+]			double rotary reflection	{01234}	[2p+,2+,2+,2+]+ = [p]+
6	[2p+,2+,2+,2+,2+]			triple rotation	{012345}	[2p+,2+,2+,2+,2+]+ = [p]+
7	[2p+,2+,2+,2+,2+,2+]			triple rotary reflection	{0123456}	[2p+,2+,2+,2+,2+,2+]+ = [p]+
4	[2p+,2+,2q+]	2pq		double rotation	{0123, 0132}	[2p+,2+,2q+]+	Double rotation: [2p+,2+,2q+]+ order pq
5	[2p+,2+,2q+,2+]			double rotary reflection	{01234, 01243}	[2p+,2+,2q+,2+]+
6	[2p+,2+,2q+,2+,2+]			triple rotation	{012345, 012354, 013245}	[2p+,2+,2q+,2+,2+]+
7	[2p+,2+,2q+,2+,2+,2+]			triple rotary reflection	{0123456, 0123465, 0124356, 0124356}	[2p+,2+,2q+,2+,2+,2+]+
6	[2p+,2+,2q+,2+,2r+]	2pqr		triple rotation	{012345, 012354, 013245}	[2p+,2+,2q+,2+,2r+]+	Triple rotation: [2p+,2+,2q+,2+,2r+]+ order pqr
7	[2p+,2+,2q+,2+,2r+,2+]			triple rotary reflection	{0123456, 0123465, 0124356, 0213456}	[2p+,2+,2q+,2+,2r+,2+]+

Commutator subgroups

Simple groups with only odd-order branch elements have only a single rotational/translational subgroup of order 2, which is also the commutator subgroup, examples [3,3]⁺, [3,5]⁺, [3,3,3]⁺, [3,3,5]⁺. For other Coxeter groups with even-order branches, the commutator subgroup has index 2^c, where c is the number of disconnected subgraphs when all the even-order branches are removed.^[6]

For example, [4,4] has three independent nodes in the Coxeter diagram when the 4s are removed, so its commutator subgroup is index 2³, and can have different representations, all with three ⁺ operators: [4⁺,4⁺]⁺, [1⁺,4,1⁺,4,1⁺], [1⁺,4,4,1⁺]⁺, or [(4⁺,4⁺,2⁺)]. A general notation can be used with +c as a group exponent, like [4,4]⁺³.

Example subgroups

Rank 2 example subgroups

Dihedral symmetry groups with even-orders have a number of subgroups. This example shows two generator mirrors of [4] in red and green, and looks at all subgroups by halfing, rank-reduction, and their direct subgroups. The group [4], has two mirror generators 0, and 1. Each generate two virtual mirrors 101 and 010 by reflection across the other.

Subgroups of [4]
Index	1	2 (half)		4 (Rank-reduction)
Diagram
Coxeter	[1,4,1] = [4]	= = [1+,4,1] = [1+,4] = [2]	= = [1,4,1+] = [4,1+] = [2]	[2,1+] = [1] = [ ]	[1+,2] = [1] = [ ]
Generators	{0,1}	{101,1}	{0,010}	{0}	{1}
Direct subgroups
Index	2	4		8
Diagram
Coxeter	[4]+	= = = [4]+2 = [1+,4,1+] = [2]+		[ ]+
Generators	{01}	{(01)2}		{02} = {12} = {(01)4} = { }

Rank 3 Euclidean example subgroups

The [4,4] group has 15 small index subgroups. This table shows them all, with a yellow fundamental domain for pure reflective groups, and alternating white and blue domains which are paired up to make rotational domains. Cyan, red, and green mirror lines correspond to the same colored nodes in the Coxeter diagram. Subgroup generators can be expressed as products of the original 3 mirrors of the fundamental domain, {0,1,2}, corresponding to the 3 nodes of the Coxeter diagram, . A product of two intersecting reflection lines makes a rotation, like {012}, {12}, or {02}. Removing a mirror causes two copies of neighboring mirrors, across the removed mirror, like {010}, and {212}. Two rotations in series cut the rotation order in half, like {0101} or {(01)²}, {1212} or {(02)²}. A product of all three mirrors creates a transreflection, like {012} or {120}.

Small index subgroups of [4,4]
Index	1	2			4
Diagram
Coxeter	[1,4,1,4,1] = [4,4]	[1+,4,4] =	[4,4,1+] =	[4,1+,4] =	[1+,4,4,1+] =	[4+,4+]
Generators	{0,1,2}	{010,1,2}	{0,1,212}	{0,101,121,2}	{010,1,212,20102}	{(01)2,(12)2,012,120}
Orbifold	*442			*2222		22×
Semidirect subgroups
Index		2			4
Diagram
Coxeter		[4,4+]	[4+,4]	[(4,4,2+)] =	[4,1+,4,1+] = =	[1+,4,1+,4] = =
Generators		{0,12}	{01,2}	{02,1,212}	{0,101,(12)2}	{(01)2,121,2}
Orbifold		4*2		2*22
Direct subgroups
Index	2	4			8
Diagram
Coxeter	[4,4]+ =	[4,4+]+ =	[4+,4]+ =	[(4,4,2+)]+ =	[4,4]+3 = [(4+,4+,2+)] = [1+,4,1+,4,1+] = [4+,4+]+ = = = =
Generators	{01,12}	{(01)2,12}	{01,(12)2}	{02,(01)2,(12)2}	{(01)2,(12)2,2(01)22}
Orbifold	442			2222
Radical subgroups
Index		8		16
Diagram
Coxeter		[4,4*] =	[4*,4] =	[4,4*]+ =	[4*,4]+ =
Orbifold		*2222		2222

Hyperbolic example subgroups

The same set of 15 small subgroups exists on all triangle groups with even order elements, like [6,4] in the hyperbolic plane:

Small index subgroups of [6,4]
Index	1	2			4
Diagram
Coxeter	[1,6,1,4,1] = [6,4]	[1+,6,4] =	[6,4,1+] =	[6,1+,4] =	[1+,6,4,1+] =	[6+,4+]
Generators	{0,1,2}	{010,1,2}	{0,1,212}	{0,101,121,2}	{010,1,212,20102}	{(01)2,(12)2,012}
Orbifold	*642	*443	*662	*3222	*3232	32×
Semidirect subgroups
Diagram
Coxeter		[6,4+]	[6+,4]	[(6,4,2+)]	[6,1+,4,1+] = = = =	[1+,6,1+,4] = = = =
Generators		{0,12}	{01,2}	{02,1,212}	{0,101,(12)2}	{(01)2,121,2}
Orbifold		4*3	6*2	2*32	2*33	3*22
Direct subgroups
Index	2	4			8
Diagram
Coxeter	[6,4]+ =	[6,4+]+ =	[6+,4]+ =	[(6,4,2+)]+ =	[6+,4+]+ = [1+,6,1+,4,1+] = = =
Generators	{01,12}	{(01)2,12}	{01,(12)2}	{02,(01)2,(12)2}	{(01)2,(12)2,201012}
Orbifold	642	443	662	3222	3232
Radical subgroups
Index		8	12	16	24
Diagram
Generators		{0,101,21012,1210121}	{2,121,101020101,0102010, 010101020101010, 10101010201010101}
Coxeter (orbifold)		[6,4*] = (*3333)	[6*,4] (*222222)	[6,4*]+ = (3333)	[6*,4]+ (222222)

Extended symmetry

Wallpaper group Triangle symmetry Extended symmetry Extended diagram Extended group Honeycombs p3m1 (*333) a1 [3[3]] ${\displaystyle {\tilde {A}}_{2}}$ (none) p6m (*632) i2 [[3[3]]] ↔ [6,3] ↔ ${\displaystyle {\tilde {A}}_{2}\times 2\leftrightarrow {\tilde {G}}_{2}}$ 1, 2 p31m (3*3) g3 [3+[3[3]]] ↔ [6,3+] ↔ ${\displaystyle {\tilde {A}}_{2}\times 3\leftrightarrow {\tfrac {1}{2}}{\tilde {G}}_{2}}$ (none) p6 (632) r6 [3[3[3]]]+ ↔ [6,3]+ ↔ ${\displaystyle {\tfrac {1}{2}}{\tilde {A}}_{2}\times 6\leftrightarrow {\tfrac {1}{2}}{\tilde {G}}_{2}}$ (1) p6m (*632) [3[3[3]]] ↔ [6,3] ${\displaystyle {\tilde {A}}_{2}\times 6\leftrightarrow {\tilde {G}}_{2}}$ 3

In the Euclidean plane, the ${\displaystyle {\tilde {A}}_{2}}$, [3[3]] Coxeter group can be extended in two ways into the ${\displaystyle {\tilde {G}}_{2}}$, [6,3] Coxeter group and relates uniform tilings as ringed diagrams.

Coxeter's notation includes double square bracket notation, [[X]] to express automorphic symmetry within a Coxeter diagram. Johnson added alternative doubling by angled-bracket <[X]>. Johnson also added a prefix symmetry modifier [Y[X]], where Y can either represent symmetry of the Coxeter diagram of [X], or symmetry of the fundamental domain of [X].

For example, in 3D these equivalent rectangle and rhombic geometry diagrams of ${\displaystyle {\tilde {A}}_{3}}$: and , the first doubled with square brackets, [[3^[4]]] or twice doubled as [2[3^[4]]], with [2], order 4 higher symmetry. To differentiate the second, angled brackets are used for doubling, <[3^[4]]> and twice doubled as <2[3^[4]]>, also with a different [2], order 4 symmetry. Finally a full symmetry where all 4 nodes are equivalent can be represented by [4[3^[4]]], with the order 8, [4] symmetry of the square. But by considering the tetragonal disphenoid fundamental domain the [4] extended symmetry of the square graph can be marked more explicitly as [(2⁺,4)[3^[4]]] or [2⁺,4[3^[4]]].

Further symmetry exists in the cyclic ${\displaystyle {\tilde {A}}_{n}}$ and branching ${\displaystyle D_{3}}$, ${\displaystyle {\tilde {E}}_{6}}$, and ${\displaystyle {\tilde {D}}_{4}}$ diagrams. ${\displaystyle {\tilde {A}}_{n}}$ has order 2n symmetry of a regular n-gon, {n}, and is represented by [n[3^[n]]]. ${\displaystyle D_{3}}$ and ${\displaystyle {\tilde {E}}_{6}}$ are represented by [3[3^1,1,1]] = [3,4,3] and [3[3^2,2,2]] respectively while ${\displaystyle {\tilde {D}}_{4}}$ by [(3,3)[3^1,1,1,1]] = [3,3,4,3], with the diagram containing the order 24 symmetry of the regular tetrahedron, {3,3}. The paracompact hyperbolic group ${\displaystyle {\bar {L}}_{5}}$ = [3^1,1,1,1,1], , contains the symmetry of a 5-cell, {3,3,3}, and thus is represented by [(3,3,3)[3^1,1,1,1,1]] = [3,4,3,3,3].

An asterisk * superscript is effectively an inverse operation, creating radical subgroups removing connected of odd-ordered mirrors.^[7]

Examples:

Example Extended groups and radical subgroups

Extended groups Radical subgroups Coxeter diagrams Index [3[2,2]] = [4,3] [4,3*] = [2,2] = 6 [(3,3)[2,2,2]] = [4,3,3] [4,(3,3)*] = [2,2,2] = 24 [1[31,1]] = [[3,3]] = [3,4] [3,4,1+] = [3,3] = 2 [3[31,1,1]] = [3,4,3] [3*,4,3] = [31,1,1] = 6 [2[31,1,1,1]] = [4,3,3,4] [1+,4,3,3,4,1+] = [31,1,1,1] = 4 [3[3,31,1,1]] = [3,3,4,3] [3*,4,3,3] = [31,1,1,1] = 6 [(3,3)[31,1,1,1]] = [3,4,3,3] [3,4,(3,3)*] = [31,1,1,1] = 24 [2[3,31,1,1,1]] = [3,(3,4)1,1] [3,(3,4,1+)1,1] = [3,31,1,1,1] = 4 [(2,3)[1,131,1,1]] = [4,3,3,4,3] [3*,4,3,3,4,1+] = [31,1,1,1,1] = 12 [(3,3)[3,31,1,1,1]] = [3,3,4,3,3] [3,3,4,(3,3)*] = [31,1,1,1,1] = 24 [(3,3,3)[31,1,1,1,1]] = [3,4,3,3,3] [3,4,(3,3,3)*] = [31,1,1,1,1] = 120

Extended groups Radical subgroups Coxeter diagrams Index [1[3[3]]] = [3,6] [3,6,1+] = [3[3]] = 2 [3[3[3]]] = [6,3] [6,3*] = [3[3]] = 6 [1[3,3[3]]] = [3,3,6] [3,3,6,1+] = [3,3[3]] = 2 [(3,3)[3[3,3]]] = [6,3,3] [6,(3,3)*] = [3[3,3]] = 24 [1[∞]2] = [4,4] [4,1+,4] = [∞]2 = [∞,2,∞] = 2 [2[∞]2] = [4,4] [1+,4,4,1+] = [(4,4,2*)] = [∞]2 = 4 [4[∞]2] = [4,4] [4,4*] = [∞]2 = 8 [2[3[4]]] = [4,3,4] [1+,4,3,4,1+] = [(4,3,4,2*)] = [3[4]] = = 4 [3[∞]3] = [4,3,4] [4,3*,4] = [∞]3 = [∞,2,∞,2,∞] = 6 [(3,3)[∞]3] = [4,31,1] [4,(31,1)*] = [∞]3 = 24 [(4,3)[∞]3] = [4,3,4] [4,(3,4)*] = [∞]3 = 48 [(3,3)[∞]4] = [4,3,3,4] [4,(3,3)*,4] = [∞]4 = 24 [(4,3,3)[∞]4] = [4,3,3,4] [4,(3,3,4)*] = [∞]4 = 384

Looking at generators, the double symmetry is seen as adding a new operator that maps symmetric positions in the Coxeter diagram, making some original generators redundant. For 3D space groups, and 4D point groups, Coxeter defines an index two subgroup of [[X]], [[X]⁺], which he defines as the product of the original generators of [X] by the doubling generator. This looks similar to [[X]]⁺, which is the chiral subgroup of [[X]]. So for example the 3D space groups [[4,3,4]]⁺ (I432, 211) and [[4,3,4]⁺] (Pm3n, 223) are distinct subgroups of [[4,3,4]] (Im3m, 229).

Rank one groups

In one dimension, the bilateral group [ ] represents a single mirror symmetry, abstract Dih₁ or Z₂, symmetry order 2. It is represented as a Coxeter–Dynkin diagram with a single node, . The identity group is the direct subgroup [ ]⁺, Z₁, symmetry order 1. The + superscript simply implies that alternate mirror reflections are ignored, leaving the identity group in this simplest case. Coxeter used a single open node to represent an alternation, .

Group	Coxeter notation	Coxeter diagram	Order	Description
C1	[ ]+		1	Identity
D2	[ ]		2	Reflection group

Rank two groups

In two dimensions, the rectangular group [2], abstract D₂² or D₄, also can be represented as a direct product [ ]×[ ], being the product of two bilateral groups, represents two orthogonal mirrors, with Coxeter diagram, , with order 4. The 2 in [2] comes from linearization of the orthogonal subgraphs in the Coxeter diagram, as with explicit branch order 2. The rhombic group, [2]⁺ ( or ), half of the rectangular group, the point reflection symmetry, Z₂, order 2.

Coxeter notation to allow a 1 place-holder for lower rank groups, so [1] is the same as [ ], and [1⁺] or [1]⁺ is the same as [ ]⁺ and Coxeter diagram .

The full p-gonal group [p], abstract dihedral group D_2p, (nonabelian for p>2), of order 2p, is generated by two mirrors at angle π/p, represented by Coxeter diagram . The p-gonal subgroup [p]⁺, cyclic group Z_p, of order p, generated by a rotation angle of π/p.

Coxeter notation uses double-bracking to represent an automorphic doubling of symmetry by adding a bisecting mirror to the fundamental domain. For example, [[p]] adds a bisecting mirror to [p], and is isomorphic to [2p].

In the limit, going down to one dimensions, the full apeirogonal group is obtained when the angle goes to zero, so [∞], abstractly the infinite dihedral group D_∞, represents two parallel mirrors and has a Coxeter diagram . The apeirogonal group [∞]⁺, , abstractly the infinite cyclic group Z_∞, isomorphic to the additive group of the integers, is generated by a single nonzero translation.

In the hyperbolic plane, there is a full pseudogonal group [iπ/λ], and pseudogonal subgroup [iπ/λ]⁺, . These groups exist in regular infinite-sided polygons, with edge length λ. The mirrors are all orthogonal to a single line.

Example rank 2 finite and hyperbolic symmetries
Type	Finite					Affine	Hyperbolic
Geometry					...
Coxeter	[ ]	= [2]=[ ]×[ ]	[3]	[4]	[p]	[∞]	[∞]	[iπ/λ]
Order	2	4	6	8	2p	∞
Mirror lines are colored to correspond to Coxeter diagram nodes. Fundamental domains are alternately colored.
Even images (direct)					...
Odd images (inverted)
Coxeter	[ ]+	[2]+	[3]+	[4]+	[p]+	[∞]+	[∞]+	[iπ/λ]+
Order	1	2	3	4	p	∞
Cyclic subgroups represent alternate reflections, all even (direct) images.

Group	Intl	Orbifold	Coxeter	Coxeter diagram	Order	Description
Finite
Zn	n	n•	[n]+		n	Cyclic: n-fold rotations. Abstract group Zn, the group of integers under addition modulo n.
D2n	nm	*n•	[n]		2n	Dihedral: cyclic with reflections. Abstract group Dihn, the dihedral group.
Affine
Z∞	∞	∞•	[∞]+		∞	Cyclic: apeirogonal group. Abstract group Z∞, the group of integers under addition.
Dih∞	∞m	*∞•	[∞]		∞	Dihedral: parallel reflections. Abstract infinite dihedral group Dih∞.
Hyperbolic
Z∞			[πi/λ]+		∞	pseudogonal group
Dih∞			[πi/λ]		∞	full pseudogonal group

Rank three groups

Point groups in 3 dimensions can be expressed in bracket notation related to the rank 3 Coxeter groups:

Finite groups of isometries in 3-space[2]
Rotation groups						Extended groups
Name	Bracket	Orb	Sch	Abstract	Order	Name	Bracket	Orb	Sch	Abstract	Order
Identity	[ ]+	11	C1	Z1	1	Bilateral	[1,1] = [ ]	*	D2	Z2	2
						Central	[2+,2+]	×	Ci	2×Z1	2
Acrorhombic	[1,2]+ = [2]+	22	C2	Z2	2	Acrorectangular	[1,2] = [2]	*22	C2v	D4	4
						Gyrorhombic	[2+,4+]	2×	S4	Z4	4
						Orthorhombic	[2,2+]	2*	D1d	Z2×Z2	4
Pararhombic	[2,2]+	222	D2	D4	4	Gyrorectangular	[2+,4]	2*2	D2d	D8	8
						Orthorectangular	[2,2]	*222	D2h	Z2×D4	8
Acro-p-gonal	[1,p]+ = [p]+	pp	Cp	Zp	p	Full acro-p-gonal	[1,p] = [p]	*pp	Cpv	D2p	2p
						Gyro-p-gonal	[2+,2p+]	p×	S2p	Z2p	2p
						Ortho-p-gonal	[2,p+]	p*	Cph	Z2×Zp	2p
Para-p-gonal	[2,p]+	p22	D2p	D2p	2p	Full gyro-p-gonal	[2+,2p]	2*p	Dpd	D4p	4p
						Full ortho-p-gonal	[2,p]	*p22	Dph	Z2×D2p	4p
Tetrahedral	[3,3]+	332	T	A4	12	Full tetrahedral	[3,3]	*332	Td	S4	24
						Pyritohedral	[3+,4]	3*2	Th	2×A4	24
Octahedral	[3,4]+	432	O	S4	24	Full octahedral	[3,4]	*432	Oh	2×S4	48
Icosahedral	[3,5]+	532	I	A5	60	Full icosahedral	[3,5]	*532	Ih	2×A5	120

In three dimensions, the full orthorhombic group or orthorectangular [2,2], abstractly Z₂³, order 8, represents three orthogonal mirrors, (also represented by Coxeter diagram as three separate dots ). It can also can be represented as a direct product [ ]×[ ]×[ ], but the [2,2] expression allows subgroups to be defined:

First there is a "semidirect" subgroup, the orthorhombic group, [2,2⁺] ( or ), abstractly Z₂×Z₂, of order 4. When the + superscript is given inside of the brackets, it means reflections generated only from the adjacent mirrors (as defined by the Coxeter diagram, ) are alternated. In general, the branch orders neighboring the + node must be even. In this case [2,2⁺] and [2⁺,2] represent two isomorphic subgroups that are geometrically distinct. The other subgroups are the pararhombic group [2,2]⁺ ( or ), also order 4, and finally the central group [2⁺,2⁺] ( or ) of order 2.

Next there is the full ortho-p-gonal group, [2,p] (), abstractly Z₂×D_2p, of order 4p, representing two mirrors at a dihedral angle π/p, and both are orthogonal to a third mirror. It is also represented by Coxeter diagram as .

The direct subgroup is called the para-p-gonal group, [2,p]⁺ ( or ), abstractly D_2p, of order 2p, and another subgroup is [2,p⁺] () abstractly Z₂×Z_p, also of order 2p.

The full gyro-p-gonal group, [2⁺,2p] ( or ), abstractly D_4p, of order 4p. The gyro-p-gonal group, [2⁺,2p⁺] ( or ), abstractly Z_2p, of order 2p is a subgroup of both [2⁺,2p] and [2,2p⁺].

The polyhedral groups are based on the symmetry of platonic solids: the tetrahedron, octahedron, cube, icosahedron, and dodecahedron, with Schläfli symbols {3,3}, {3,4}, {4,3}, {3,5}, and {5,3} respectively. The Coxeter groups for these are: [3,3] (), [3,4] (), [3,5] () called full tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, with orders of 24, 48, and 120.

In all these symmetries, alternate reflections can be removed producing the rotational tetrahedral [3,3]⁺(), octahedral [3,4]⁺ (), and icosahedral [3,5]⁺ () groups of order 12, 24, and 60. The octahedral group also has a unique index 2 subgroup called the pyritohedral symmetry group, [3⁺,4] ( or ), of order 12, with a mixture of rotational and reflectional symmetry. Pyritohedral symmetry is also an index 5 subgroup of icosahedral symmetry: --> , with virtual mirror 1 across 0, {010}, and 3-fold rotation {12}.

The tetrahedral group, [3,3] (), has a doubling [[3,3]] (which can be represented by colored nodes ), mapping the first and last mirrors onto each other, and this produces the [3,4] ( or ) group. The subgroup [3,4,1⁺] ( or ) is the same as [3,3], and [3⁺,4,1⁺] ( or ) is the same as [3,3]⁺.

Example rank 3 finite Coxeter groups subgroup trees
Tetrahedral symmetry	Octahedral symmetry
Icosahedral symmetry

Finite (point groups in three dimensions)

Intl* Geo [8] Orb. Schön. Struct. Coxeter Ord. 1 1 1 C1 Z1 [] = [ ]+ = 1 2 = m 1 * Cs Z2 [ ] 2 2 3 4 5 6 n 2 3 4 5 6 n 22 33 44 55 66 nn C2 C3 C4 C5 C6 Cn Z2 Z3 Z4 Z5 Z6 Zn [1,2]+ [1,3]+ [1,4]+ [1,5]+ [1,6]+ [1,n]+ 2 3 4 5 6 n 2mm 3m 4mm 5m 6mm nmm nm 2 3 4 5 6 n *22 *33 *44 *55 *66 *nn C2v C3v C4v C5v C6v Cnv D2 D3 D4 D5 D6 Dn [1,2] [1,3] [1,4] [1,5] [1,6] [1,n] 4 6 8 10 12 2n 2/m 3/m 4/m 5/m 6/m n/m 2 2 3 2 4 2 5 2 6 2 n 2 2* 3* 4* 5* 6* n* C2h C3h C4h C5h C6h Cnh Z2×Z2 Z2×Z3 Z2×Z4 Z2×Z5 Z2×Z6 Z2×Zn [2,2+] [2,3+] [2,4+] [2,5+] [2,6+] [2,n+] 4 6 8 10 12 2n 1 4 3 8 5 12 2n n 2 2 4 2 6 2 8 2 10 2 12 2 2n 2 1× 2× 3× 4× 5× 6× n× Ci = S2 S4 S6 S8 S10 S12 S2n Z2 Z4 Z6=Z2×Z3 Z8 Z10=Z2×Z5 Z12 Z2n [2+,2+] [2+,4+] [2+,6+] [2+,8+] [2+,10+] [2+,12+] [2+,2n+] 2 4 6 8 10 12 2n

Intl Geo Orb. Schön. Struct. Coxeter Ord. 222 32 422 52 622 n22 n2 2 2 3 2 4 2 5 2 6 2 n 2 222 223 224 225 226 22n D2 D3 D4 D5 D6 Dn D4 D6 D8 D10 D12 D2n [2,2]+ [2,3]+ [2,4]+ [2,5]+ [2,6]+ [2,n]+ 4 6 8 10 12 2n mmm 6m2 4/mmm 10m2 6/mmm n/mmm 2nm2 2 2 3 2 4 2 5 2 6 2 n 2 *222 *223 *224 *225 *226 *22n D2h D3h D4h D5h D6h Dnh Z2×D4 Z2×D6 Z2×D8 Z2×D10 Z2×D12 Z2×D2n [2,2] [2,3] [2,4] [2,5] [2,6] [2,n] 8 12 16 20 24 4n 42m 3m 82m 5m 122m 2n2m nm 4 2 6 2 8 2 10 2 12 2 n 2 2*2 2*3 2*4 2*5 2*6 2*n D2d D3d D4d D5d D6d Dnd D4 D6 D8 D10 D12 D2n [2+,4] [2+,6] [2+,8] [2+,10] [2+,12] [2+,2n] 8 12 16 20 24 4n 23 3 3 332 T A4 [3,3]+ 12 m3 4 3 3*2 Th A4×S2 [3+,4] 24 43m 3 3 *332 Td S4 [3,3] 24 432 4 3 432 O S4 [3,4]+ 24 m3m 4 3 *432 Oh S4×S2 [3,4] 48 532 5 3 532 I A5 [3,5]+ 60 53m 5 3 *532 Ih A4×S2 [3,5] 120

Affine

In the Euclidean plane there's 3 fundamental reflective groups generated by 3 mirrors, represented by Coxeter diagrams , , and , and are given Coxeter notation as [4,4], [6,3], and [(3,3,3)]. The parentheses of the last group imply the diagram cycle, and also has a shorthand notation [3^[3]].

[[4,4]] as a doubling of the [4,4] group produced the same symmetry rotated π/4 from the original set of mirrors.

Direct subgroups of rotational symmetry are: [4,4]⁺, [6,3]⁺, and [(3,3,3)]⁺. [4⁺,4] and [6,3⁺] are semidirect subgroups.

Semiaffine (frieze groups) IUC Orb. Geo Sch. Coxeter p1 ∞∞ p1 C∞ [∞] = [∞,1]+ = [∞+,2,1+] = p1m1 *∞∞ p1 C∞v [∞] = [∞,1] = [∞,2,1+] = p11g ∞× p.g1 S2∞ [∞+,2+] p11m ∞* p. 1 C∞h [∞+,2] p2 22∞ p2 D∞ [∞,2]+ p2mg 2*∞ p2g D∞d [∞,2+] p2mm *22∞ p2 D∞h [∞,2]

Affine (Wallpaper groups) IUC Orb. Geo. Coxeter p2 2222 p2 [4,1+,4]+ p2gg 22× pg2g [4+,4+] p2mm *2222 p2 [4,1+,4] c2mm 2*22 c2 [[4+,4+]] p4 442 p4 [4,4]+ p4gm 4*2 pg4 [4+,4] p4mm *442 p4 [4,4] p3 333 p3 [1+,6,3+] = [3[3]]+ = p3m1 *333 p3 [1+,6,3] = [3[3]] = p31m 3*3 h3 [6,3+] = [3[3[3]]+] p6 632 p6 [6,3]+ = [3[3[3]]]+ p6mm *632 p6 [6,3] = [3[3[3]]]

Given in Coxeter notation (orbifold notation), some low index affine subgroups are:

Reflective group	Reflective subgroup	Mixed subgroup	Rotation subgroup	Improper rotation/ translation	Commutator subgroup
[4,4], (*442)	[1+,4,4], (*442) [4,1+,4], (*2222) [1+,4,4,1+], (*2222)	[4+,4], (4*2) [(4,4,2+)], (2*22) [1+,4,1+,4], (2*22)	[4,4]+, (442) [1+,4,4+], (442) [1+,4,1+4,1+], (2222)	[4+,4+], (22×)	[4+,4+]+, (2222)
[6,3], (*632)	[1+,6,3] = [3[3]], (*333)	[3+,6], (3*3)	[6,3]+, (632) [1+,6,3+], (333)		[1+,6,3+], (333)

Rank four groups

Subgroup relations

Point groups

Rank four groups defined the 4-dimensional point groups:

Finite groups
[ ]: Symbol Order [1]+ 1.1 [1] = [ ] 2.1 [2]: Symbol Order [1+,2]+ 1.1 [2]+ 2.1 [2] 4.1 [2,2]: Symbol Order [2+,2+]+ = [(2+,2+,2+)] 1.1 [2+,2+] 2.1 [2,2]+ 4.1 [2+,2] 4.1 [2,2] 8.1 [2,2,2]: Symbol Order [(2+,2+,2+,2+)] = [2+,2+,2+]+ 1.1 [2+,2+,2+] 2.1 [2+,2,2+] 4.1 [(2,2)+,2+] 4 [[2+,2+,2+]] 4 [2,2,2]+ 8 [2+,2,2] 8.1 [(2,2)+,2] 8 [[2+,2,2+]] 8.1 [2,2,2] 16.1 [[2,2,2]]+ 16 [[2,2+,2]] 16 [[2,2,2]] 32	[p]: Symbol Order [p]+ p [p] 2p [p,2]: Symbol Order [p,2]+ 2p [p,2] 4p [2p,2+]: Symbol Order [2p,2+] 4p [2p+,2+] 2p [p,2,2]: Symbol Order [p+,2,2+] 2p [(p,2)+,2+] 2p [p,2,2]+ 4p [p,2,2+] 4p [p+,2,2] 4p [(p,2)+,2] 4p [p,2,2] 8p [2p,2+,2]: Symbol Order [2p+,2+,2+]+ p [2p+,2+,2+] 2p [2p+,2+,2] 4p [2p+,(2,2)+] 4p [2p,(2,2)+] 8p [2p,2+,2] 8p	[p,2,q]: Symbol Order [p+,2,q+] pq [p,2,q]+ 2pq [p+,2,q] 2pq [p,2,q] 4pq [(p,2)+,2q]: Symbol Order [(p,2)+,2q+] 2pq [(p,2)+,2q] 4pq [2p,2,2q]: Symbol Order [2p+,2+,2q+]+= [(2p+,2+,2q+,2+)] pq [2p+,2+,2q+] 2pq [2p,2+,2q+] 4pq [((2p,2)+,(2q,2)+)] 4pq [2p,2+,2q] 8pq [[p,2,p]]: Symbol Order [[p+,2,p+]] 2p2 [[p,2,p]]+ 4p2 [[p,2,p]+] 4p2 [[p,2,p]] 8p2 [[2p,2,2p]]: Symbol Order [[(2p+,2+,2p+,2+)]] 2p2 [[2p+,2+,2p+]] 4p2 [[((2p,2)+,(2p,2)+)]] 8p2 [[2p,2+,2p]] 16p2	[3,3,2]: Symbol Order [(3,3)Δ,2,1+] ≅ [2,2]+ 4 [(3,3)Δ,2] ≅ [2,(2,2)+] 8 [(3,3)⅄,2,1+] ≅ [4,2+] 8 [(3,3)+,2,1+] = [3,3]+ 12.5 [(3,3)⅄,2] ≅ [2,4,2+] 16 [3,3,2,1+] = [3,3] 24 [(3,3)+,2] 24.10 [3,3,2]+ 24.10 [3,3,2] 48.36 [4,3,2]: Symbol Order [1+,4,3+,2,1+] = [3,3]+ 12 [3+,4,2+] 24 [(3,4)+,2+] 24 [1+,4,3+,2] = [(3,3)+,2] 24.10 [3+,4,2,1+] = [3+,4] 24.10 [(4,3)+,2,1+] = [4,3]+ 24.15 [1+,4,3,2,1+] = [3,3] 24 [1+,4,(3,2)+] = [3,3,2]+ 24 [3,4,2+] 48 [4,3+,2] 48.22 [4,(3,2)+] 48 [(4,3)+,2] 48.36 [1+,4,3,2] = [3,3,2] 48.36 [4,3,2,1+] = [4,3] 48.36 [4,3,2]+ 48.36 [4,3,2] 96.5 [5,3,2]: Symbol Order [(5,3)+,2,1+] = [5,3]+ 60.13 [5,3,2,1+] = [5,3] 120.2 [(5,3)+,2] 120.2 [5,3,2]+ 120.2 [5,3,2] 240 (nc)	[31,1,1]: Symbol Order [31,1,1]Δ ≅[[4,2+,4]]+ 32 [31,1,1]⅄ 64 [31,1,1]+ 96.1 [31,1,1] 192.2 <[3,31,1]> = [4,3,3] 384.1 [3[31,1,1]] = [3,4,3] 1152.1 [3,3,3]: Symbol Order [3,3,3]+ 60.13 [3,3,3] 120.1 [[3,3,3]]+ 120.2 [[3,3,3]+] 120.1 [[3,3,3]] 240.1 [4,3,3]: Symbol Order [1+,4,(3,3)Δ] = [31,1,1]Δ ≅[[4,2+,4]]+ 32 [4,(3,3)Δ] = [2+,4[2,2,2]+] ≅[[4,2+,4]] 64 [1+,4,(3,3)⅄] = [31,1,1]⅄ 64 [1+,4,(3,3)+] = [31,1,1]+ 96.1 [4,(3,3)⅄] ≅ [[4,2,4]] 128 [1+,4,3,3] = [31,1,1] 192.2 [4,(3,3)+] 192.1 [4,3,3]+ 192.3 [4,3,3] 384.1 [3,4,3]: Symbol Order [3+,4,3+] 288.1 [3,4,3⅄] = [4,3,3] 384.1 [3,4,3]+ 576.2 [3+,4,3] 576.1 [[3+,4,3+]] 576 (nc) [3,4,3] 1152.1 [[3,4,3]]+ 1152 (nc) [[3,4,3]+] 1152 (nc) [[3,4,3]] 2304 (nc) [5,3,3]: Symbol Order [5,3,3]+ 7200 (nc) [5,3,3] 14400 (nc)

Subgroups

1D-4D reflective point groups and subgroups
Order	Reflection		Semidirect subgroups				Direct subgroups		Commutator subgroup
2	[ ]						[ ]+		[ ]+1	[ ]+
4	[2]						[2]+		[2]+2
8	[2,2]		[2+,2]		[2+,2+]		[2,2]+		[2,2]+3
16	[2,2,2]		[2+,2,2] [(2,2)+,2]		[2+,2+,2] [(2,2)+,2+] [2+,2+,2+]		[2,2,2]+ [2+,2,2+]		[2,2,2]+4
	[21,1,1]				[(2+)1,1,1]
2n	[n]						[n]+		[n]+1	[n]+
4n	[2n]						[2n]+		[2n]+2
4n	[2,n]		[2,n+]				[2,n]+		[2,n]+2
8n	[2,2n]		[2+,2n]		[2+,2n+]		[2,2n]+		[2,2n]+3
8n	[2,2,n]		[2+,2,n] [2,2,n+]		[2+,(2,n)+]		[2,2,n]+ [2+,2,n+]		[2,2,n]+3
16n	[2,2,2n]		[2,2+,2n]		[2+,2+,2n] [2,2+,2n+] [(2,2)+,2n+] [2+,2+,2n+]		[2,2,2n]+ [2+,2n,2+]		[2,2,2n]+4
	[2,2n,2]				[2+,2n+,2+]
	[2n,21,1]				[2n+,(2+)1,1]
24	[3,3]						[3,3]+		[3,3]+1	[3,3]+
48	[3,3,2]		[(3,3)+,2]				[3,3,2]+		[3,3,2]+2
48	[4,3]		[4,3+]				[4,3]+		[4,3]+2
96	[4,3,2]		[(4,3)+,2] [4,(3,2)+]				[4,3,2]+		[4,3,2]+3
	[3,4,2]		[3,4,2+] [3+,4,2]		[(3,4)+,2+]		[3+,4,2+]
120	[5,3]						[5,3]+		[5,3]+1	[5,3]+
240	[5,3,2]		[(5,3)+,2]				[5,3,2]+		[5,3,2]+2
4pq	[p,2,q]		[p+,2,q]				[p,2,q]+ [p+,2,q+]		[p,2,q]+2	[p+,2,q+]
8pq	[2p,2,q]		[2p,(2,q)+]		[2p+,(2,q)+]		[2p,2,q]+		[2p,2,q]+3
16pq	[2p,2,2q]		[2p,2+,2q]		[2p+,2+,2q] [2p+,2+,2q+] [(2p,(2,2q)+,2+)]	-	[2p,2,2q]+		[2p,2,2q]+4
120	[3,3,3]						[3,3,3]+		[3,3,3]+1	[3,3,3]+
192	[31,1,1]						[31,1,1]+		[31,1,1]+1	[31,1,1]+
384	[4,3,3]		[4,(3,3)+]				[4,3,3]+		[4,3,3]+2
1152	[3,4,3]		[3+,4,3]				[3,4,3]+ [3+,4,3+]		[3,4,3]+2	[3+,4,3+]
14400	[5,3,3]						[5,3,3]+		[5,3,3]+1	[5,3,3]+

Space groups

Space groups
Affine isomorphism and correspondences	8 cubic space groups as extended symmetry from [3[4]], with square Coxeter diagrams and reflective fundamental domains	35 cubic space groups in International, Fibrifold notation, and Coxeter notation

Rank four groups as 3-dimensional space groups
Triclinic (1-2) Coxeter Space group [∞+,2,∞+,2,∞+] (1) P1 Monoclinic (3-15) Coxeter Space group [(∞,2,∞)+,2,∞+] (3) P2 [∞+,2,∞+,2,∞] (6) Pm [(∞,2,∞)+,2,∞] (10) P2/m Orthorhombic (16-74) Coxeter Space group [∞,2,∞,2,∞]+ (16) P222 [[∞,2,∞,2,∞]]+ (23) I222 [∞+,2,∞,2,∞] (25) Pmm2 [∞,2,∞,2,∞] (47) Pmmm [[∞,2,∞,2,∞]] (71) Immm [∞+,2,∞+,2,∞+] [∞,2,∞,2+,∞] [∞,2+,∞,2+,∞]	Tetragonal (75-142) Coxeter Space group [(4,4)+,2,∞+] (75) P4 [2+[(4,4)+,2,∞+]] (79) I4 [(4,4)+,2,∞] (83) P4/m [2+[(4,4)+,2,∞]] (87) I4/m [4,4,2,∞]+ (89) P422 [2+[4,4,2,∞]]+ (97) I422 [4,4,2,∞+] (99) P4mm [4,4,2,∞] (123) P4/mmm [2+[4,4,2,∞]] (139) I4/mmm [4,(4,2)+,∞] (140) I4/mcm [4,4,2+,∞] [(4,4)+,2+,∞] [4,4,2+,∞+] [(4,4)+,2+,∞+] [4+,4+,2+,∞] [4,4+,2,∞] [4,4+,2+,∞] [((4,2+,4)),2,∞] [4,4+,2,∞+] [4,4+,2+,∞+] [((4,2+,4)),2,∞+]	Trigonal (143-167), rhombohedral Coxeter Space group Hexagonal (168-194) [(6,3)+,2,∞+] (168) P6 [(6,3)+,2,∞] (175) P6/m [6,3,2,∞]+ (177) P622 [6,3,2,∞+] (183) P6mm [6,3,2,∞] (191) P6/mmm [(3[3])+,2,∞+] [3[3],2,∞] [6,3+,2,∞] [6,3+,2,∞+] [3[3],2,∞]+ [3[3],2,∞+] [(3[3])+,2,∞]	Cubic (195-230) Group Coxeter Space group Index [[4,3,4]] [[4,3,4]] (229) Im3m 1 [[4,3,4]]+ (211) I432 2 [[4,3,4]+] (223) Pm3n 2 [[4,3+,4]] (204) I3 2 [[(4,3,4,2+)]] (217) I43m 2 [[4,3+,4]]+ (197) I23 4 [[4,3,4]+]+ (208) P4232 4 [[4,3+,4)]+] (201) Pn43 4 [[(4,3,4,2+)]+] (218) P43n 4 [4,3,4] [4,3,4] (221) Pm3m 2 [4,3,4]+ (207) P432 4 [4,3+,4] (200) Pm3 4 [4,(3,4)+] (226) Fm3c 4 [(4,3,4,2+)] (215) P43m 4 [[{4,(3}+,4)+]] (228) Fd3c 4 [4,3+,4]+ (195) P23 8 [{4,(3}+,4)+] (219) F43c 8 [4,31,1] [4,31,1] (225) Fm3m 4 [4,(31,1)+] (202) Fm3 8 [4,31,1]+ (209) F432 8 [[3[4]]] [(4+,2+)[3[4]]] (222) Pn3n 2 [[3[4]]] (227) Fd3m 4 [[3[4]]]+ (203) Fd3 8 [[3[4]]+] (210) F4132 8 [3[4]] [3[4]] (216) F43m 8 [3[4]]+ (196) F23 16

Line groups

Rank four groups also defined the 3-dimensional line groups:

Semiaffine (3D) groups
Point group			Line group
Hermann-Mauguin		Schönflies	Hermann-Mauguin		Offset type	Wallpaper			Coxeter [∞h,2,pv]
Even n	Odd n		Even n	Odd n		IUC	Orbifold	Diagram
n		Cn	Pnq		Helical: q	p1	o		[∞+,2,n+]
2n	n	S2n	P2n	Pn	None	p11g, pg(h)	××		[(∞,2)+,2n+]
n/m	2n	Cnh	Pn/m	P2n	None	p11m, pm(h)	**		[∞+,2,n]
2n/m		C2nh	P2nn/m		Zigzag	c11m, cm(h)	*×		[∞+,2+,2n]
nmm	nm	Cnv	Pnmm	Pnm	None	p1m1, pm(v)	**		[∞,2,n+]
			Pncc	Pnc	Planar reflection	p1g1, pg(v)	××		[∞+,(2,n)+]
2nmm		C2nv	P2nnmc		Zigzag	c1m1, cm(v)	*×		[∞,2+,2n+]
n22	n2	Dn	Pnq22	Pnq2	Helical: q	p2	2222		[∞,2,n]+
2n2m	nm	Dnd	P2n2m	Pnm	None	p2mg, pmg(h)	22*		[(∞,2)+,2n]
			P2n2c	Pnc	Planar reflection	p2gg, pgg	22×		[+(∞,(2),2n)+]
n/mmm	2n2m	Dnh	Pn/mmm	P2n2m	None	p2mm, pmm	*2222		[∞,2,n]
			Pn/mcc	P2n2c	Planar reflection	p2mg, pmg(v)	22*		[∞,(2,n)+]
2n/mmm		D2nh	P2nn/mcm		Zigzag	c2mm, cmm	2*22		[∞,2+,2n]

Duoprismatic group

Extended duoprismatic symmetry
Extended duoprismatic groups, [p]×[p] or [p,2,p] or , expressed in relation to its tetragonal disphenoid fundamental domain symmetry.

Rank four groups defined the 4-dimensional duoprismatic groups. In the limit as p and q go to infinity, they degenerate into 2 dimensions and the wallpaper groups.

Duoprismatic groups (4D)
Wallpaper			Coxeter [p,2,q]	Coxeter [[p,2,p]]	Wallpaper
IUC	Orbifold	Diagram			IUC	Orbifold	Diagram
p1	o		[p+,2,q+]	[[p+,2,p+]]	p1	o
pg	××		[(p,2)+,2q+]	-
pm	**		[p+,2,q]	-
cm	*×		[2p+,2+,2q]	-
p2	2222		[p,2,q]+	[[p,2,p]]+	p4	442
pmg	22*		[(p,2)+,2q]	-
pgg	22×		[+(2p,(2),2q)+]	[[+(2p,(2),2p)+]]	cmm	2*22
pmm	*2222		[p,2,q]	[[p,2,p]]	p4m	*442
cmm	2*22		[2p,2+,2q]	[[2p,2+,2p]]	p4g	4*2

Wallpaper groups

Rank four groups also defined some of the 2-dimensional wallpaper groups, as limiting cases of the four-dimensional duoprism groups:

Affine (2D plane)
IUC Orb. Geo Coxeter Diagram p1 o p1 [∞+,2,∞+] [∞+,2+,∞+] [(∞+,2+,∞+,2+)] p2 2222 p2 [∞,2,∞]+ [(∞,2+,∞,2+)] p11g ×× pg1 h: [∞+,(2,∞)+] p1g1 v: [(∞,2)+,∞+] p2gm 22* pg2 h: [(∞,2)+,∞] p2mg v: [∞,(2,∞)+]	IUC Orb. Geo Coxeter Diagram p11m ** p1 h: [∞+,2,∞] p1m1 v: [∞,2,∞+] p2mm *2222 p2 [∞,2,∞] c11m *× c1 h: [∞+,2+,∞] c1m1 v: [∞,2+,∞+] p2gg 22× pg2g [+(∞,(2),∞)+] [((∞,2)+)[2]] c2mm 2*22 c2 [∞,2+,∞]

Subgroups of [∞,2,∞], (*2222) can be expressed down to its index 16 commutator subgroup:

Subgroups of [∞,2,∞]
Reflective group	Reflective subgroup	Mixed subgroup	Rotation subgroup	Improper rotation/ translation	Commutator subgroup
[∞,2,∞], (*2222)	[1+,∞,2,∞], (*2222)	[∞+,2,∞], (**)	[∞,2,∞]+, (2222)	[∞,2+,∞]+, (°) [∞+,2+,∞+], (°) [∞+,2,∞+], (°) [∞+,2+,∞], (*×) [(∞,2)+,∞+], (××) [((∞,2)+,(∞,2)+)], (22×)	[(∞+,2+,∞+,2+)], (°)
		[∞,2+,∞], (2*22) [(∞,2)+,∞], (22*)

Complex reflections

Coxeter notation has been extended to Complex space, Cⁿ where nodes are unitary reflections of period 2 or greater. Nodes are labeled by an index, assumed to be 2 for ordinary real reflection if suppressed. Complex reflection groups are called Shephard groups rather than Coxeter groups, and can be used to construct complex polytopes.

In ${\displaystyle \mathbb {C} ^{1}}$, a rank 1 shephard group , order p, is represented as _p[ ], [ ]_p or ]_p[. It has a single generator, representing a 2π/p radian rotation in the Complex plane: ${\displaystyle e^{2\pi i/p}}$.

Coxeter writes the rank 2 complex group, _p[q]_r represents Coxeter diagram . The p and r should only be suppressed if both are 2, which is the real case [q]. The order of a rank 2 group _p[q]_r is ${\displaystyle g=8/q(1/p+2/q+1/r-1)^{-2}}$.^[9]

The rank 2 solutions that generate complex polygons are: _p[4]₂ (p is 2,3,4,...), ₃[3]₃, ₃[6]₂, ₃[4]₃, ₄[3]₄, ₃[8]₂, ₄[6]₂, ₄[4]₃, ₃[5]₃, ₅[3]₅, ₃[10]₂, ₅[6]₂, and ₅[4]₃ with Coxeter diagrams , , , , , , , , , , , , .

Infinite groups are ₃[12]₂, ₄[8]₂, ₆[6]₂, ₃[6]₃, ₆[4]₃, ₄[4]₄, and ₆[3]₆ or , , , , , , .

Index 2 subgroups exists by removing a real reflection: _p[2q]₂ → _p[q]_p. Also index r subgroups exist for 4 branches: _p[4]_r → _p[r]_p.

For the infinite family _p[4]₂, for any p = 2, 3, 4,..., there are two subgroups: _p[4]₂ → [p], index p, while and _p[4]₂ → _p[ ]×_p[ ], index 2.

Computation with reflection matrices as symmetry generators

A Coxeter group, represented by Coxeter diagram , is given Coxeter notation [p,q] for the branch orders. Each node in the Coxeter diagram represents a mirror, by convention called ρ_i (and matrix R_i). The generators of this group [p,q] are reflections: ρ₀, ρ₁, and ρ₂. Rotational subsymmetry is given as products of reflections: By convention, σ_0,1 (and matrix S_0,1) = ρ₀ρ₁ represents a rotation of angle π/p, and σ_1,2 = ρ₁ρ₂ is a rotation of angle π/q, and σ_0,2 = ρ₀ρ₂ represents a rotation of angle π/2.

[p,q]⁺, , is an index 2 subgroup represented by two rotation generators, each a products of two reflections: σ_0,1, σ_1,2, and representing rotations of π/p, and π/q angles respectively.

With one even branch, [p⁺,2q], or , is another subgroup of index 2, represented by rotation generator σ_0,1, and reflectional ρ₂.

With even branches, [2p⁺,2q⁺], , is a subgroup of index 4 with two generators, constructed as a product of all three reflection matrices: By convention as: ψ_0,1,2 and ψ_1,2,0, which are rotary reflections, representing a reflection and rotation or reflection.

In the case of affine Coxeter groups like , or , one mirror, usually the last, is translated off the origin. A translation generator τ_0,1 (and matrix T_0,1) is constructed as the product of two (or an even number of) reflections, including the affine reflection. A transreflection (reflection plus a translation) can be the product of an odd number of reflections φ_0,1,2 (and matrix V_0,1,2), like the index 4 subgroup : [4⁺,4⁺] = .

Another composite generator, by convention as ζ (and matrix Z), represents the inversion, mapping a point to its inverse. For [4,3] and [5,3], ζ = (ρ₀ρ₁ρ₂)^h/2, where h is 6 and 10 respectively, the Coxeter number for each family. For 3D Coxeter group [p,q] (), this subgroup is a rotary reflection [2⁺,h⁺].

Coxeter groups are categorized by their rank, being the number of nodes in its Coxeter-Dynkin diagram. The structure of the groups are also given with their abstract group types: In this article, the abstract dihedral groups are represented as Dih_n, and cyclic groups are represented by Z_n, with Dih₁=Z₂.

Rank 2

Dihedral groups	Cyclic groups
[2]	[2]+
[3]	[3]+
[4]	[4]+
[6]	[6]+

Example, in 2D, the Coxeter group [p] () is represented by two reflection matrices R₀ and R₁, The cyclic symmetry [p]⁺ () is represented by rotation generator of matrix S_0,1.

[p], Reflections Rotation Name R0 R1 S0,1=R0×R1 Order 2 2 p Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0\\0&-1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p\\\sin 2\pi /p&-\cos 2\pi /p\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p\\-\sin 2\pi /p&\cos 2\pi /p\\\end{smallmatrix}}\right]}$	[2], Reflections Rotation Name R0 R1 S0,1=R0×R1 Order 2 2 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0\\0&-1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1&0\\0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1&0\\0&-1\\\end{smallmatrix}}\right]}$
[3], Reflections Rotation Name R0 R1 S0,1=R0×R1 Order 2 2 3 Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0\\0&-1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1/2&{\sqrt {3}}/2\\{\sqrt {3}}/2&-1/2\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1/2&{\sqrt {3}}/2\\-{\sqrt {3}}/2&1/2\\\end{smallmatrix}}\right]}$	[4], Reflections Rotation Name R0 R1 S0,1=R0×R1 Order 2 2 4 Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0\\0&-1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1\\-1&0\\\end{smallmatrix}}\right]}$
[6], Reflections Rotation Name R0 R1 S0,1=R0×R1 Order 2 2 6 Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0\\0&-1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}{\sqrt {3}}/2&1/2\\1/2&-{\sqrt {3}}/2\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}{\sqrt {3}}/2&1/2\\-1/2&{\sqrt {3}}/2\\\end{smallmatrix}}\right]}$	[8], Reflections Rotation Name R0 R1 S0,1=R0×R1 Order 2 2 8 Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0\\0&-1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}{\sqrt {2}}/2&{\sqrt {2}}/2\\{\sqrt {2}}/2&-{\sqrt {2}}/2\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}{\sqrt {2}}/2&{\sqrt {2}}/2\\-{\sqrt {2}}/2&{\sqrt {2}}/2\\\end{smallmatrix}}\right]}$

Rank 3

The finite rank 3 Coxeter groups are [1,p], [2,p], [3,3], [3,4], and [3,5].

To reflect a point through a plane ${\displaystyle ax+by+cz=0}$ (which goes through the origin), one can use ${\displaystyle \mathbf {A} =\mathbf {I} -2\mathbf {NN} ^{T}}$, where ${\displaystyle \mathbf {I} }$ is the 3×3 identity matrix and ${\displaystyle \mathbf {N} }$ is the three-dimensional unit vector for the vector normal of the plane. If the L2 norm of ${\displaystyle a,b,}$ and ${\displaystyle c}$ is unity, the transformation matrix can be expressed as:

${\displaystyle \mathbf {A} =\left[{\begin{smallmatrix}1-2a^{2}&-2ab&-2ac\\-2ab&1-2b^{2}&-2bc\\-2ac&-2bc&1-2c^{2}\end{smallmatrix}}\right]}$

[p,2]

The reducible 3-dimensional finite reflective group is dihedral symmetry, [p,2], order 4p, . The reflection generators are matrices R₀, R₁, R₂. R₀²=R₁²=R₂²=(R₀×R₁)³=(R₁×R₂)³=(R₀×R₂)²=Identity. [p,2]⁺ () is generated by 2 of 3 rotations: S_0,1, S_1,2, and S_0,2. An order p rotoreflection is generated by V_0,1,2, the product of all 3 reflections.

[p,2],

	Reflections			Rotation			Rotoreflection
Name	R0	R1	R2	S0,1	S1,2	S0,2	V0,1,2
Group
Order	2	2	2	p	2		2p
Matrix	${\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p&0\\\sin 2\pi /p&-\cos 2\pi /p&0\\0&0&1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&1&0\\0&0&-1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p&0\\-\sin 2\pi /p&\cos 2\pi /p&0\\0&0&1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p&0\\-\sin 2\pi /p&\cos 2\pi /p&0\\0&0&-1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&-1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&-\sin 2\pi /p&0\\-\sin 2\pi /p&-\cos 2\pi /p&0\\0&0&1\\\end{smallmatrix}}\right]}$

[3,3]

The simplest irreducible 3-dimensional finite reflective group is tetrahedral symmetry, [3,3], order 24, . The reflection generators, from a D₃=A₃ construction, are matrices R₀, R₁, R₂. R₀²=R₁²=R₂²=(R₀×R₁)³=(R₁×R₂)³=(R₀×R₂)²=Identity. [3,3]⁺ () is generated by 2 of 3 rotations: S_0,1, S_1,2, and S_0,2. A trionic subgroup, isomorphic to [2⁺,4], order 8, is generated by S_0,2 and R₁. An order 4 rotoreflection is generated by V_0,1,2, the product of all 3 reflections.

[3,3],

	Reflections			Rotations			Rotoreflection
Name	R0	R1	R2	S0,1	S1,2	S0,2	V0,1,2
Name
Order	2	2	2	3		2	4
Matrix	${\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&0&1\\0&1&0\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}0&1&0\\1&0&0\\0&0&1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&0&-1\\0&-1&0\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}0&1&0\\0&0&1\\1&0&0\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}0&0&-1\\1&0&0\\0&-1&0\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&-1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}0&0&-1\\0&-1&0\\1&0&0\\\end{smallmatrix}}\right]}$
	(0,1,−1)n	(1,−1,0)n	(0,1,1)n	(1,1,1)axis	(1,1,−1)axis	(1,0,0)axis

[4,3]

Another irreducible 3-dimensional finite reflective group is octahedral symmetry, [4,3], order 48, . The reflection generators matrices are R₀, R₁, R₂. R₀²=R₁²=R₂²=(R₀×R₁)⁴=(R₁×R₂)³=(R₀×R₂)²=Identity. Chiral octahedral symmetry, [4,3]⁺, () is generated by 2 of 3 rotations: S_0,1, S_1,2, and S_0,2. Pyritohedral symmetry [4,3⁺], () is generated by reflection R₀ and rotation S_1,2. A 6-fold rotoreflection is generated by V_0,1,2, the product of all 3 reflections.

[4,3],

	Reflections			Rotations			Rotoreflection
Name	R0	R1	R2	S0,1	S1,2	S0,2	V0,1,2
Group
Order	2	2	2	4	3	2	6
Matrix	${\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&1&0\\0&0&-1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&0&1\\0&1&0\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}0&1&0\\1&0&0\\0&0&1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&0&1\\0&-1&0\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}0&1&0\\0&0&1\\1&0&0\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}0&1&0\\1&0&0\\0&0&-1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}0&1&0\\0&0&1\\-1&0&0\\\end{smallmatrix}}\right]}$
	(0,0,1)n	(0,1,−1)n	(1,−1,0)n	(1,0,0)axis	(1,1,1)axis	(1,−1,0)axis

[5,3]

A final irreducible 3-dimensional finite reflective group is icosahedral symmetry, [5,3], order 120, . The reflection generators matrices are R₀, R₁, R₂. R₀²=R₁²=R₂²=(R₀×R₁)⁵=(R₁×R₂)³=(R₀×R₂)²=Identity. [5,3]⁺ () is generated by 2 of 3 rotations: S_0,1, S_1,2, and S_0,2. A 10-fold rotoreflection is generated by V_0,1,2, the product of all 3 reflections.

[5,3],

	Reflections			Rotations			Rotoreflection
Name	R0	R1	R2	S0,1	S1,2	S0,2	V0,1,2
Group
Order	2	2	2	5	3	2	10
Matrix	${\displaystyle \left[{\begin{smallmatrix}-1&0&0\\0&1&0\\0&0&1\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {-\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&1\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}-1&0&0\\0&-1&0\\0&0&1\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {-\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}$
	(1,0,0)n	(φ,1,φ−1)n	(0,1,0)n	(φ,1,0)axis	(1,1,1)axis	(1,0,0)axis

Rank 4

There are 4 irreducible Coxeter groups in 4 dimensions: [3,3,3], [4,3,3], [3^1,1,1], [3,4,4], [5,3,3], as well as an infinite family of duoprismatic groups [p,2,q].

[p,2,q]

The duprismatic group, [p,2,q], has order 4pq.

[p,2,q], Reflections Name R0 R1 R2 R3 Group element Order 2 2 2 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p&0&0\\\sin 2\pi /p&-\cos 2\pi /p&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos 2\pi /q&\sin 2\pi /q\\0&0&\sin 2\pi /q&-\cos 2\pi /q\\\end{smallmatrix}}\right]}$ [[p,2,p]] The duoprismatic group can double in order, to 8p2, with a 2-fold rotation between the two planes. [[p,2,p]], Rotation Reflections Name T R0 R1 R2=TR1T R3=TR0T Element Order 2 2 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}\cos 2\pi /p&\sin 2\pi /p&0&0\\\sin 2\pi /p&-\cos 2\pi /p&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&0&0&-1\\0&0&-1&0\\0&-1&0&0\\-1&0&0&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos 2\pi /p&\sin 2\pi /p\\0&0&\sin 2\pi /p&-\cos 2\pi /p\\\end{smallmatrix}}\right]}$ [3,3,3] Hypertetrahedral symmetry, [3,3,3], order 120, is easiest to represent with 4 mirrors in 5-dimensions, as a subgroup of [4,3,3,3]. [3,3,3], Reflections Rotations Rotoreflections Double rotation Name R0 R1 R2 R3 S0,1 S1,2 S2,3 S0,2 S1,3 S2,3 V0,1,2 V0,1,3 W0,1,2,3 Name Order 2 2 2 2 3 2 4 6 5 Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&1\\0&0&0&1&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&0&1&0\\0&0&1&0&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&0&1&0&0\\0&1&0&0&0\\0&0&0&1&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0\\1&0&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&1&0&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&1&0&0&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&1&0&0&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&0&1&0&0\\0&1&0&0&0\\0&0&0&0&1\\0&0&0&1&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0\\1&0&0&0&0\\0&0&0&1&0\\0&0&1&0&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0\\1&0&0&0&0\\0&0&1&0&0\\0&0&0&0&1\\0&0&0&1&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&1&0&0&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0\\1&0&0&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&1&0&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\1&0&0&0&0\\\end{smallmatrix}}\right]}$ (0,0,0,1,-1)n (0,0,1,−1,0)n (0,1,−1,0,0)n (1,−1,0,0,0)n [[3,3,3]] The extended group [[3,3,3]], order 240, is doubled by a 2-fold rotation matrix T, here reversing coordinate order and sign: There are 3 generators {T, R0, R1}. Since T is self-reciprocal R3=TR0T, and R2=TR1T. [[3,3,3]], Rotation Reflections Name T R0 R1 TR1T=R2 TR0T=R3 Element group Order 2 2 2 2 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}0&0&0&0&-1\\0&0&0&-1&0\\0&0&-1&0&0\\0&-1&0&0&0\\-1&0&0&0&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&1\\0&0&0&1&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&0&1&0\\0&0&1&0&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0\\0&0&1&0&0\\0&1&0&0&0\\0&0&0&1&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0\\1&0&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\\end{smallmatrix}}\right]}$ (0,0,0,1,-1)n (0,0,1,−1,0)n (0,1,−1,0,0)n (1,−1,0,0,0)n [4,3,3] A irreducible 4-dimensional finite reflective group is hyperoctahedral group (or hexadecachoric group (for 16-cell), B4=[4,3,3], order 384, . The reflection generators matrices are R0, R1, R2, R3. R02=R12=R22=R32=(R0×R1)4=(R1×R2)3=(R2×R3)3=(R0×R2)2=(R1×R3)2=(R0×R3)2=Identity. Chiral hyperoctahedral symmetry, [4,3,3]+, () is generated by 3 of 6 rotations: S0,1, S1,2, S2,3, S0,2, S1,3, and S0,3. Hyperpyritohedral symmetry [4,(3,3)+], () is generated by reflection R0 and rotations S1,2 and S2,3. An 8-fold double rotation is generated by W0,1,2,3, the product of all 4 reflections. [4,3,3], Reflections Rotations Rotoreflection Double rotation Name R0 R1 R2 R3 S0,1 S1,2 S2,3 S0,2 S1,3 S0,3 V1,2,3 V0,1,3 V0,1,2 V0,2,3 W0,1,2,3 Group Order 2 2 2 2 4 3 2 4 6 8 Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&0&0&1\\0&1&0&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&-1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&1&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&-1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&-1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\-1&0&0&0\\\end{smallmatrix}}\right]}$ (0,0,0,1)n (0,0,1,−1)n (0,1,−1,0)n (1,−1,0,0)n [3,31,1] A half group of [4,3,3] is [3,31,1], , order 192. It shares 3 generators with [4,3,3] group, but has two copies of an adjacent generator, one reflected across the removed mirror. [3,31,1], Reflections Name R0 R1 R2 R3 Group Order 2 2 2 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&-1\\0&0&-1&0\\\end{smallmatrix}}\right]}$ (1,−1,0,0)n (0,1,−1,0)n (0,0,1,−1)n (0,0,1,1)n [3,4,3] A irreducible 4-dimensional finite reflective group is Icositetrachoric group (for 24-cell), F4=[3,4,3], order 1152, . The reflection generators matrices are R0, R1, R2, R3. R02=R12=R22=R32=(R0×R1)3=(R1×R2)4=(R2×R3)3=(R0×R2)2=(R1×R3)2=(R0×R3)2=Identity. Chiral icositetrachoric symmetry, [3,4,3]+, () is generated by 3 of 6 rotations: S0,1, S1,2, S2,3, S0,2, S1,3, and S0,3. Ionic diminished [3,4,3+] group, () is generated by reflection R0 and rotations S1,2 and S2,3. A 12-fold double rotation is generated by W0,1,2,3, the product of all 4 reflections. [3,4,3], Reflections Rotations Name R0 R1 R2 R3 S0,1 S1,2 S2,3 S0,2 S1,3 S0,3 Element group Order 2 2 2 2 3 4 3 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1/2&-1/2&-1/2&-1/2\\-1/2&1/2&-1/2&-1/2\\-1/2&-1/2&1/2&-1/2\\-1/2&-1/2&-1/2&1/2\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1/2&-1/2&1/2&-1/2\\-1/2&1/2&1/2&-1/2\\-1/2&-1/2&-1/2&-1/2\\-1/2&-1/2&1/2&1/2\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1/2&-1/2&-1/2&-1/2\\-1/2&-1/2&1/2&-1/2\\-1/2&1/2&-1/2&-1/2\\-1/2&-1/2&-1/2&1/2\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1/2&1/2&-1/2&-1/2\\1/2&-1/2&-1/2&-1/2\\-1/2&-1/2&1/2&-1/2\\-1/2&1/2&-1/2&1/2\\\end{smallmatrix}}\right]}$ (1,−1,0,0)n (0,1,−1,0)n (0,0,1,0)n (−1,−1,−1,−1)n [3,4,3], Rotoreflection Double rotation Name V1,2,3 V0,1,3 V0,1,2 V0,2,3 W0,1,2,3 Element group Order 6 12 Matrix ${\displaystyle \left[{\begin{smallmatrix}-1/2&1/2&-1/2&-1/2\\-1/2&-1/2&1/2&-1/2\\1/2&-1/2&-1/2&-1/2\\-1/2&-1/2&-1/2&1/2\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\-1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1/2&1/2&-1/2&-1/2\\-1/2&1/2&1/2&-1/2\\-1/2&-1/2&-1/2&-1/2\\-1/2&1/2&-1/2&1/2\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1/2&1/2&1/2&-1/2\\1/2&-1/2&1/2&-1/2\\-1/2&-1/2&-1/2&-1/2\\-1/2&-1/2&1/2&1/2\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1/2&1/2&-1/2&-1/2\\1/2&-1/2&1/2&-1/2\\-1/2&-1/2&-1/2&-1/2\\1/2&-1/2&-1/2&1/2\\\end{smallmatrix}}\right]}$ [[3,4,3]] The group [[3,4,3]] extends [3,4,3] by a 2-fold rotation, T, doubling order to 2304. [[3,4,3]], Rotation Reflections Name T R0 R1 R2 = TR1T R3 = TR0T Element group Order 2 2 2 2 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1/2&-1/2&-1/2&-1/2\\-1/2&1/2&-1/2&-1/2\\-1/2&-1/2&1/2&-1/2\\-1/2&-1/2&-1/2&1/2\\\end{smallmatrix}}\right]}$ (1,−1,0,0)n (0,1,−1,0)n (0,0,1,0)n (−1,−1,−1,−1)n [5,3,3] Stereographic projections [5,3,3]+ 72 order-5 gyrations [5,3,3]+ 200 order-3 gyrations [5,3,3]+ 450 order-2 gyrations [5,3,3]+ all gyrations The hyper-icosahedral symmetry, [5,3,3], order 14400, . The reflection generators matrices are R0, R1, R2, R3. R02=R12=R22=R32=(R0×R1)5=(R1×R2)3=(R2×R3)3=(R0×R2)2=(R0×R3)2=(R1×R3)2=Identity. [5,3,3]+ () is generated by 3 rotations: S0,1 = R0×R1, S1,2 = R1×R2, S2,3 = R2×R3, etc. [5,3,3], Reflections Name R0 R1 R2 R3 Element group Order 2 2 2 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {-\phi }{2}}&{\frac {-1}{2}}&0\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}&0\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}&0\\0&0&0&1\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&1\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&{\frac {1}{2}}&{\frac {\phi }{2}}&{\frac {1-\phi }{2}}\\0&{\frac {\phi }{2}}&{\frac {1-\phi }{2}}&{\frac {1}{2}}\\0&{\frac {1-\phi }{2}}&{\frac {1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]}$ (1,0,0,0)n (φ,1,φ−1,0)n (0,1,0,0)n (0,−1,φ,1−φ)n Rank 8 [34,2,1] The E8 Coxeter group, [34,2,1], , has 8 mirror nodes, order 696729600 (192x10!). E7 and E6, [33,2,1], , and [32,2,1], can be constructed by ignoring the first mirror or the first two mirrors respectively. E8=[34,2,1], Reflections Name R0 R1 R2 R3 R4 R5 R6 R7 Element group Order 2 2 2 2 2 2 2 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0&0&0&0&0\\1&0&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&0&-1&0\\0&0&0&0&0&-1&0&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}3/4&-1/4&-1/4&-1/4&-1/4&-1/4&-1/4&-1/4\\-1/4&3/4&-1/4&-1/4&-1/4&-1/4&-1/4&-1/4\\-1/4&-1/4&3/4&-1/4&-1/4&-1/4&-1/4&-1/4\\-1/4&-1/4&-1/4&3/4&-1/4&-1/4&-1/4&-1/4\\-1/4&-1/4&-1/4&-1/4&3/4&-1/4&-1/4&-1/4\\-1/4&-1/4&-1/4&-1/4&-1/4&3/4&-1/4&-1/4\\-1/4&-1/4&-1/4&-1/4&-1/4&-1/4&3/4&-1/4\\-1/4&-1/4&-1/4&-1/4&-1/4&-1/4&-1/4&3/4\end{smallmatrix}}\right]}$ (1,-1,0,0,0,0,0,0)n (0,1,-1,0,0,0,0,0)n (0,0,1,-1,0,0,0,0)n (0,0,0,1,-1,0,0,0)n (0,0,0,0,1,-1,0,0)n (0,0,0,0,0,1,-1,0)n (0,0,0,0,0,1,1,0)n (1,1,1,1,1,1,1,1)n Affine rank 2 Affine matrices are represented by adding an extra row and column, the last row being zero except last entry 1. The last column represents a translation vector. [∞] The affine group [∞], , can be given by two reflection matrices, x=0 and x=1. [∞], Reflections Translation Name R0 R1 S0,1 Element group Order 2 2 ∞ Matrix ${\displaystyle \left[{\begin{smallmatrix}-1&0\\0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1&1\\0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&-1\\0&1\\\end{smallmatrix}}\right]}$ Hyperplane x=0 x=1 Affine rank 3 [4,4] Further information: Tetrakis_square_tiling § Symmetry The affine group [4,4], , (p4m), can be given by three reflection matrices, reflections across the x axis (y=0), a diagonal (x=y), and the affine reflection across the line (x=1). [4,4]+ () (p4) is generated by S0,1 S1,2, and S0,2. [4+,4+] () (pgg) is generated by 2-fold rotation S0,2 and glide reflection (transreflection) V0,1,2. [4+,4] () (p4g) is generated by S0,1 and R3. The group [(4,4,2+)] () (cmm), is generated by 2-fold rotation S1,3 and reflection R2. [4,4], Reflections Rotations Glides Name R0 R1 R2 S0,1 S1,2 S0,2 V0,1,2 V0,2,1 Element group Order 2 2 2 4 2 ∞ (2) Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0\\1&0&0\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1&0&2\\0&1&0\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0\\-1&0&0\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0\\-1&0&2\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1&0&2\\0&-1&0\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0\\1&0&-2\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&-1&2\\-1&0&0\\0&0&1\\\end{smallmatrix}}\right]}$ Hyperplane y=0 x=y x=1 [3,6] The affine group [3,6], , (p6m), can be given by three reflection matrices, reflections across the x axis (y=0), line y=(√3/2)x, and vertical line x=1. [3,6], Reflections Rotations Glides Name R0 R1 R2 S0,1 S1,2 S0,2 V0,1,2 V0,2,1 Element group Order 2 2 2 3 6 2 ∞ (2) Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1/2&{\sqrt {3}}/2&0\\{\sqrt {3}}/2&1/2&0\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1&0&2\\0&1&0\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1/2&{\sqrt {3}}/2&0\\-{\sqrt {3}}/2&-1/2&0\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1/2&{\sqrt {3}}/2&-1\\-{\sqrt {3}}/2&1/2&{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1&0&2\\0&-1&0\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1/2&{\sqrt {3}}/2&-1\\{\sqrt {3}}/2&-1/2&-{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1/2&-{\sqrt {3}}/2&2\\-{\sqrt {3}}/2&-1/2&0\\0&0&1\\\end{smallmatrix}}\right]}$ Hyperplane y=0 y=(√3/2)x x=1 [3[3]] The affine group [3[3]] can be constructed as a half group of . R2 is replaced by R'2 = R2×R1×R2, presented by the hyperplane: y+(√3/2)x=2. The fundamental domain is an equilateral triangle with edge length 2. [3[3]], Reflections Rotations Glides Name R0 R1 R'2 = R2×R1×R2 S0,1 S1,2 S0,2 V0,1,2 V0,2,1 Element group Order 2 2 2 3 ∞ (2) Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1/2&{\sqrt {3}}/2&0\\{\sqrt {3}}/2&1/2&0\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1/2&-{\sqrt {3}}/2&3\\-{\sqrt {3}}/2&1/2&{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0\\-1&0&0\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1/2&{\sqrt {3}}/2&0\\-{\sqrt {3}}/2&-1/2&2{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1/2&-{\sqrt {3}}/2&3\\{\sqrt {3}}/2&-1/2&-{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1/2&{\sqrt {3}}/2&0\\{\sqrt {3}}/2&1/2&-2{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1/2&-{\sqrt {3}}/2&3\\-{\sqrt {3}}/2&1/2&-{\sqrt {3}}\\0&0&1\\\end{smallmatrix}}\right]}$ Hyperplane y=0 y=(√3/2)x y+(√3/2)x=2 Affine rank 4 [4,3,4] [4,3,4] fundamental domain The affine group is [4,3,4] (), can be given by four reflection matrices. Mirror R0 can be put on z=0 plane. Mirror R1 can be put on plane y=z. Mirror R2 can be put on x=y plane. Mirror R3 can be put on x=1 plane. [4,3,4]+ () is generated by S0,1, S1,2, and S2,3. [4,3,4], Reflections Rotations Transflections Screw axis Name R0 R1 R2 R3 S0,1 S1,2 S2,3 S0,2 S0,3 S1,3 T0,1,2 T1,2,3 U0,1,2,3 Element group Order 2 2 2 2 4 3 4 2 6 ∞ (3) Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1&0&0&2\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\-1&0&0&2\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1&0&0&2\\0&1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1&0&0&2\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\-1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\-1&0&0&2\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&-2\\0&0&0&1\\\end{smallmatrix}}\right]}$ Hyperplane z=0 y=z x=y x=1 [[4,3,4]] The extended group [[4,3,4]] doubles the group order, adding with a 2-fold rotation matrix T, with a fixed axis through points (1,1/2,0) and (1/2,1/2,1/2). The generators are {R0,R1,T}. R2 = T×R1×T and R3 = T×R0×T. [[4,3,4]], Rotation Reflections Name T R0 R1 R2 = T×R1×T R3 = T×R0×T Element group Order 2 2 2 2 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}0&0&-1&1\\0&-1&0&1\\-1&0&0&1\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1&0&0&2\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ Hyperplane Point (1/2,1/2,1/2) Axis (-1,0,1) z=0 y=z x=y x=1 [4,31,1] [4,31,1] fundamental domain The group [4,31,1] can be constructed from [4,3,4], by computing [4,3,4,1+], , as R'3=R3×R2×R3, with new R'3 as an image of R2 across R3. [4,31,1], Reflections Rotations Name R0 R1 R2 R'3 S0,1 S1,2 S1,3 S0,2 S0,3 S2,3 Element group Order 2 2 2 2 3 3 3 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&-1&0&2\\-1&0&0&2\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&-1&0&2\\0&0&1&0\\-1&0&0&2\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&-1&0&2\\-1&0&0&2\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1&0&0&2\\0&-1&0&2\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ Hyperplane z=0 y=z x=y x+y=2 [3[4]] [3[4]] fundamental domain The group [3[4]] can be constructed from [4,3,4], by removing first and last mirrors, [1+,4,3,4,1+], , by R'1=R0×R1×R0 and R'3=R3×R2×R3. [3[4]] Reflections Rotations Name R'0 R1 R2 R'3 S0,1 S1,2 S1,3 S0,2 S0,3 S2,3 Element group Order 2 2 2 2 3 3 3 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&-1&0\\0&-1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&-1&0&2\\-1&0&0&2\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&-1&0&2\\0&0&1&0\\-1&0&0&2\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1&0&0\\0&0&-1&0\\-1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&-1&0&2\\0&0&-1&0\\1&0&0&-2\\0&0&0&1\\\end{smallmatrix}}\right]}$ ${\displaystyle \left[{\begin{smallmatrix}-1&0&0&2\\0&-1&0&2\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]}$ Hyperplane y=-z y=z x=y x+y=2 Notes ^ Johnson (2018), 11.6 Subgroups and extensions, p 255, halving subgroups ^ a b Johnson (2018), pp.231-236, and p 245 Table 11.4 Finite groups of isometries in 3-space ^ Johnson (2018), 11.6 Subgroups and extensions, p 259, radical subgroup ^ Johnson (2018), 11.6 Subgroups and extensions, p 258, trionic subgroups ^ Conway, 2003, p.46, Table 4.2 Chiral groups II ^ Coxeter and Moser, 1980, Sec 9.5 Commutator subgroup, p. 124–126 ^ Johnson, Norman W.; Weiss, Asia Ivić (1999). "Quaternionic modular groups". Linear Algebra and Its Applications. 295 (1–3): 159–189. doi:10.1016/S0024-3795(99)00107-X. ^ The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF [1] ^ Coxeter, Regular Complex Polytopes, 9.7 Two-generator subgroups reflections. pp. 178–179 References H.S.M. Coxeter: Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN978-0-471-01003-6[2] (Paper 22) Coxeter, H.S.M. (1940), "Regular and Semi Regular Polytopes I", Math. Z., 46: 380–407, doi:10.1007/bf01181449, S2CID 186237114 (Paper 23) Coxeter, H.S.M. (1985), "Regular and Semi-Regular Polytopes II", Math. Z., 188 (4): 559–591, doi:10.1007/bf01161657, S2CID 120429557 (Paper 24) Coxeter, H.S.M. (1988), "Regular and Semi-Regular Polytopes III", Math. Z., 200: 3–45, doi:10.1007/bf01161745, S2CID 186237142 Coxeter, H. S. M.; Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9. Norman JohnsonUniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966) Norman W. Johnson and Asia Ivic Weiss Quadratic Integers and Coxeter Groups PDF Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336 N. W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups [3] Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P. (2001), "On three-dimensional space groups", Beiträge zur Algebra und Geometrie, 42 (2): 475–507, ISSN 0138-4821, MR 1865535 John H. Conway and Derek A. Smith, On Quaternions and Octonions, 2003, ISBN 978-1-56881-134-5 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Ch.22 35 prime space groups, ch.25 184 composite space groups, ch.26 Higher still, 4D point groups