Polytope of Type {6,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4}*720
if this polytope has a name.
Group : SmallGroup(720,767)
Rank : 3
Schlafli Type : {6,4}
Number of vertices, edges, etc : 90, 180, 60
Order of s₀s₁s₂ : 15
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {6,4,2} of size 1440
Vertex Figure Of :
   {2,6,4} of size 1440
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,4}*240c
   6-fold quotients : {6,4}*120
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,8}*1440a, {6,8}*1440b, {6,4}*1440b
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      30 facets:
         30 of {6}*12
      45 vertex figures:
         45 of {4}*8
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> 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*s0*s1> 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*s1*s2*s1*s0*s1*s2*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*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*s1*s0*s2*s1*s0*s1*s0*s2> of order 5.
      12 facets:
         12 of {6}*12
      18 vertex figures:
         18 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0> 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*s0*s1, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 12.
      7 facets:
         3 of {2}*4
         4 of {6}*12
      9 vertex figures:
         6 of {4}*8
         3 of {2}*4

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

Polytope of Type {6,4}

Twisty Puzzle