Polytope of Type {5,4} with group SmallGroup(1440,5843)

Overview

Group: SmallGroup(1440,5843)
Rank: 3
Schläfli Type: {5,4}
Vertices, edges, …: 180, 360, 144
Order of s₀s₁s₂: 8
Order of s₀s₁s₂s₁: 10
Also known as: {5,4}₈. if this polytope has another name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

2-fold

{5,4}*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=<s1*s2*(s1*s0)^2*(s2*s1*s0)^2*s1*s2> of order 2

72 facets

72 of {5}*10

92 vertex figures

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

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

48 facets

48 of {5}*10

60 vertex figures

60 of {4}*8

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

36 facets

36 of {5}*10

48 vertex figures

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

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

24 facets

24 of {5}*10

32 vertex figures

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

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

16 facets

16 of {5}*10

20 vertex figures

20 of {4}*8

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

12 facets

12 of {5}*10

16 vertex figures

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

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

8 facets

8 of {5}*10

12 vertex figures

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

Representations

Permutation Representation (GAP)

s0 := ( 1, 3)( 2, 6)( 4, 9)( 5, 7)( 8,10);;
s1 := ( 1, 2)( 3, 4)( 5,10)( 6, 7)( 8, 9);;
s2 := ( 1, 3)( 2, 5)( 4, 8)( 6, 7)( 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*s0*s1*s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(12)!( 1, 3)( 2, 6)( 4, 9)( 5, 7)( 8,10);
s1 := Sym(12)!( 1, 2)( 3, 4)( 5,10)( 6, 7)( 8, 9);
s2 := Sym(12)!( 1, 3)( 2, 5)( 4, 8)( 6, 7)( 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*s0*s1*s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References

None.

to this polytope.