Uniform 4-polytope
Symmetry of uniform 4-polytopes in four dimensions
There are 5 fundamental mirror symmetry point group families in 4-dimensions: A₄ =, B₄ =, D₄ =, F₄ =, H₄ =.

Uniform 4-polytope

Section Title: Symmetry of uniform 4-polytopes in four dimensions
Table Section Text: There are 5 fundamental mirror symmetry point group families in 4-dimensions: A₄ =, B₄ =, D₄ =, F₄ =, H₄ =.

Weyl group	Conway Quaternion	Abstract structure	Order	Coxeter diagram	Coxeter notation	Commutator subgroup	Coxeter number (h)	Mirrors m=2h

Irreducible

A4	+1/60[I×I].21	S5	120	[3,3,3]	[3,3,3]+	5	10

D4	±1/3[T×T].2	1/2.2S4	192	[31,1,1]	[31,1,1]+	6	12

B4	±1/6[O×O].2	2S4 = S2≀S4	384	[4,3,3]	8	4	12

F4	±1/2[O×O].23	3.2S4	1152	[3,4,3]	[3+,4,3+]	12	12	12

H4	±[I×I].2	2.(A5×A5).2	14400	[5,3,3]	[5,3,3]+	30	60

Prismatic groups

A3A1	+1/24[O×O].23	S4×D1	48	[3,3,2] = [3,3]×[ ]	[3,3]+	6	1

B3A1	±1/24[O×O].2	S4×D1	96	[4,3,2] = [4,3]×[ ]	3	6	1

H3A1	±1/60[I×I].2	A5×D1	240	[5,3,2] = [5,3]×[ ]	[5,3]+	15	1

Duoprismatic groups (Use 2p,2q for even integers)

I2(p)I2(q)	±1/2[D2p×D2q]	Dp×Dq	4pq	[p,2,q] = [p]×[q]	[p+,2,q+]	p	q

I2(2p)I2(q)	±1/2[D4p×D2q]	D2p×Dq	8pq	[2p,2,q] = [2p]×[q]	p	p	q

I2(2p)I2(2q)	±1/2[D4p×D4q]	D2p×D2q	16pq	[2p,2,2q] = [2p]×[2q]	p	p	q	q

Sentence(s)

Uniform 4-polytope family of order 384 with [4,3,3].

final_sentence: Uniform 4-polytope family of order 384 with [4,3,3].