Polytope of Type {6,10}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,10}*1440f
if this polytope has a name.
Group : SmallGroup(1440,5853)
Rank : 3
Schlafli Type : {6,10}
Number of vertices, edges, etc : 72, 360, 120
Order of s₀s₁s₂ : 30
Order of s₀s₁s₂s₁ : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,10}*720b, {6,10}*720b, {6,10}*720c
   3-fold quotients : {6,10}*480c
   4-fold quotients : {3,10}*360
   6-fold quotients : {3,10}*240, {6,5}*240b, {6,10}*240c, {6,10}*240d, {6,10}*240e, {6,10}*240f
   12-fold quotients : {3,5}*120, {3,10}*120a, {3,10}*120b, {6,5}*120b, {6,5}*120c
   24-fold quotients : {3,5}*60
   60-fold quotients : {6,2}*24
   120-fold quotients : {3,2}*12
   180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
      60 facets:
         60 of {6}*12
      36 vertex figures:
         36 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      60 facets:
         60 of {6}*12
      36 vertex figures:
         36 of {10}*20
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      60 facets:
         60 of {6}*12
      48 vertex figures:
         24 of {5}*10
         24 of {10}*20
   P/N, where N=<s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      60 facets:
         60 of {6}*12
      36 vertex figures:
         36 of {10}*20
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      60 facets:
         60 of {6}*12
      36 vertex figures:
         36 of {10}*20
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2> of order 3.
      40 facets:
         40 of {6}*12
      24 vertex figures:
         24 of {10}*20
   P/N, where N=<s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2, s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      30 facets:
         30 of {6}*12
      18 vertex figures:
         18 of {10}*20
   P/N, where N=<s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 4.
      30 facets:
         30 of {6}*12
      18 vertex figures:
         18 of {10}*20
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 5.
      24 facets:
         24 of {6}*12
      24 vertex figures:
         12 of {10}*20
         12 of {2}*4
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 6.
      20 facets:
         20 of {6}*12
      12 vertex figures:
         12 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2> of order 6.
      20 facets:
         20 of {6}*12
      12 vertex figures:
         12 of {10}*20
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 10.
      12 facets:
         12 of {6}*12
      12 vertex figures:
         6 of {10}*20
         6 of {2}*4
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 10.
      12 facets:
         12 of {6}*12
      12 vertex figures:
         6 of {10}*20
         6 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 12.
      10 facets:
         10 of {6}*12
      6 vertex figures:
         6 of {10}*20

Permutation Representation (GAP) :

s0 := ( 3, 6)( 4, 5)( 7,10)( 8, 9)(12,16)(13,15);;
s1 := ( 1, 3)( 2, 4)( 7,10)( 8, 9)(11,12)(13,16)(14,15);;
s2 := ( 1, 2)( 3, 8)( 4, 7)( 5,10)( 6, 9);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(16)!( 3, 6)( 4, 5)( 7,10)( 8, 9)(12,16)(13,15);
s1 := Sym(16)!( 1, 3)( 2, 4)( 7,10)( 8, 9)(11,12)(13,16)(14,15);
s2 := Sym(16)!( 1, 2)( 3, 8)( 4, 7)( 5,10)( 6, 9);
poly := sub<Sym(16)|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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 >;

References : None.
to this polytope

Polytope of Type {6,10}

Twisty Puzzle