Polytope of Type {8,4} with group SmallGroup(1440,5843)

Overview

Group: SmallGroup(1440,5843)
Rank: 3
Schläfli Type: {8,4}
Vertices, edges, …: 180, 360, 90
Order of s₀s₁s₂: 5
Order of s₀s₁s₂s₁: 10
Also known as: {8,4}₅. if this polytope has another name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable

Quotients maximal quotients in bold

2-fold

{8,4}*720

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1)^2*(s2*(s1*s0)^2*s1)^2> of order 2

46 facets

44 of {8}*16
2 of {4}*8

92 vertex figures

88 of {4}*8
4 of {2}*4

P/N, where N=<s0*s1*s2*s1*s0*s2*(s1*s0)^2*(s1*s2)^2> of order 3

30 facets

30 of {8}*16

60 vertex figures

60 of {4}*8

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, (s0*s1)^2*(s2*(s1*s0)^2*s1)^2> of order 4

24 facets

21 of {8}*16
3 of {4}*8

48 vertex figures

42 of {4}*8
6 of {2}*4

P/N, where N=<s0*s2*s1*s0*s1*(s2*s1*s0)^2*s2, s0*s1*s0*s2*(s1*s0)^3*s1*s2*s1*s0> of order 6

16 facets

14 of {8}*16
2 of {4}*8

32 vertex figures

28 of {4}*8
4 of {2}*4

P/N, where N=<(s0*s1)^2*(s2*s1*s0*s1)^2, s0*s1*s2*s1*s0*s2*(s1*s0)^2*(s1*s2)^2> of order 9

10 facets

10 of {8}*16

20 vertex figures

20 of {4}*8

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s2*(s1*s0)^2*s1*s2*s1*s0*(s1*s2)^2> of order 12

8 facets

7 of {8}*16
1 of {4}*8

16 vertex figures

14 of {4}*8
2 of {2}*4

P/N, where N=<s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, (s0*s1)^2*(s2*s1*s0*s1)^2, (s1*s0)^2*s1*(s2*s1*s0)^2*s1> of order 18

6 facets

4 of {8}*16
2 of {4}*8

12 vertex figures

8 of {4}*8
4 of {2}*4

Representations

Permutation Representation (GAP)

s0 := ( 2, 8)( 3, 6)( 4, 9)( 7,10)(11,12);;
s1 := ( 1, 2)( 3, 6)( 4, 7)( 5, 9)( 8,10);;
s2 := ( 1, 5)( 2, 3)( 4, 7)( 6, 8)( 9,10)(11,12);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(12)!( 2, 8)( 3, 6)( 4, 9)( 7,10)(11,12);
s1 := Sym(12)!( 1, 2)( 3, 6)( 4, 7)( 5, 9)( 8,10);
s2 := Sym(12)!( 1, 5)( 2, 3)( 4, 7)( 6, 8)( 9,10)(11,12);
poly := sub<Sym(12)|s0,s1,s2>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References

None.

to this polytope.