Uniform 5-polytope
Symmetry of uniform 5-polytopes in four dimensions
The 5-simplex is the regular form in the A₅ family. The 5-cube and 5-orthoplex are the regular forms in the B₅ family.

Uniform 5-polytope

Section Title: Symmetry of uniform 5-polytopes in four dimensions
Table Section Text: The 5-simplex is the regular form in the A₅ family. The 5-cube and 5-orthoplex are the regular forms in the B₅ family.

Coxeter group	Order	Coxeter diagram	Coxeter notation	Commutator subgroup	Reflections

A4A1	120	[3,3,3,2] = [3,3,3]×[ ]	[3,3,3]+	10	1

D4A1	384	[31,1,1,2] = [31,1,1]×[ ]	[31,1,1]+	12	1

B4A1	768	[4,3,3,2] = [4,3,3]×[ ]	4	12	1

F4A1	2304	[3,4,3,2] = [3,4,3]×[ ]	[3+,4,3+]	12	12	1

H4A1	28800	[5,3,3,2] = [3,4,3]×[ ]	[5,3,3]+	60	1

Duoprismatic (use 2p and 2q for evens)

I2(p)I2(q)A1	8pq	[p,2,q,2] = [p]×[q]×[ ]	[p+,2,q+]	p	q	1

I2(2p)I2(q)A1	16pq	[2p,2,q,2] = [2p]×[q]×[ ]	p	p	q	1

I2(2p)I2(2q)A1	32pq	[2p,2,2q,2] = [2p]×[2q]×[ ]	p	p	q	q	1

Sentence(s)

The A₁ x H₄ family has a symmetry of order 28800.

final_sentence: The A₁ x H₄ family has a symmetry of order 28800.