Polytope of Type {6,4} with group SmallGroup(720,767)

Overview

Group: SmallGroup(720,767)
Rank: 3
Schläfli Type: {6,4}
Vertices, edges, …: 90, 180, 60
Order of s₀s₁s₂: 15
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable

Quotients maximal quotients in bold

3-fold

{6,4}*240c

6-fold

{6,4}*120

Covers minimal covers in bold

2-fold

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*s2*s1)^3> of order 2

30 facets

30 of {6}*12

45 vertex figures

45 of {4}*8

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

30 facets

30 of {6}*12

48 vertex figures

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

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

22 facets

3 of {2}*4
19 of {6}*12

30 vertex figures

30 of {4}*8

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

15 facets

15 of {6}*12

27 vertex figures

18 of {4}*8
9 of {2}*4

P/N, where N=<s0*s1*(s2*(s1*s0)^2)^2*s2> of order 5

12 facets

12 of {6}*12

18 vertex figures

18 of {4}*8

P/N, where N=<(s0*s2*s1)^3, (s0*s1)^2*(s2*s1*s0)^2> of order 10

6 facets

6 of {6}*12

12 vertex figures

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

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

7 facets

3 of {2}*4
4 of {6}*12

9 vertex figures

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

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.