Polytope of Type {6,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*1800a
if this polytope has a name.
Group : SmallGroup(1800,575)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 150, 450, 150
Order of s0s1s2 : 30
Order of s0s1s2s1 : 6
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) :
   3-fold quotients : {6,6}*600b
   6-fold quotients : {3,6}*300
   25-fold quotients : {6,6}*72b
   50-fold quotients : {6,3}*36
   75-fold quotients : {2,6}*24
   150-fold quotients : {2,3}*12
   225-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=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      75 facets:
         75 of {6}*12
      78 vertex figures:
         72 of {6}*12
         6 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1> of order 2.
      100 facets:
         50 of {3}*6
         50 of {6}*12
      75 vertex figures:
         75 of {6}*12
   P/N, where N=<s0*s1*s0*s1> of order 3.
      54 facets:
         6 of {2}*4
         48 of {6}*12
      50 vertex figures:
         50 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 5.
      30 facets:
         30 of {6}*12
      30 vertex figures:
         30 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1> of order 5.
      30 facets:
         30 of {6}*12
      30 vertex figures:
         30 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 10.
      15 facets:
         15 of {6}*12
      18 vertex figures:
         12 of {6}*12
         6 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 10.
      20 facets:
         10 of {3}*6
         10 of {6}*12
      15 vertex figures:
         15 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2> of order 10.
      15 facets:
         15 of {6}*12
      18 vertex figures:
         6 of {3}*6
         12 of {6}*12

Permutation Representation (GAP) : 
s0 := ( 2, 5)( 3, 4)( 6, 7)( 8,10)(11,13)(14,15)(16,19)(17,18)(21,25)(22,24)(27,30)(28,29)(31,32)(33,35)(36,38)(39,40)(41,44)(42,43)(46,50)(47,49)(52,55)(53,54)(56,57)(58,60)(61,63)(64,65)(66,69)(67,68)(71,75)(72,74);;
s1 := ( 2, 7)( 3,13)( 4,19)( 5,25)( 6,21)( 9,14)(10,20)(11,16)(12,22)(18,23)(26,51)(27,57)(28,63)(29,69)(30,75)(31,71)(32,52)(33,58)(34,64)(35,70)(36,66)(37,72)(38,53)(39,59)(40,65)(41,61)(42,67)(43,73)(44,54)(45,60)(46,56)(47,62)(48,68)(49,74)(50,55);;
s2 := ( 1,37)( 2,38)( 3,39)( 4,40)( 5,36)( 6,31)( 7,32)( 8,33)( 9,34)(10,35)(11,30)(12,26)(13,27)(14,28)(15,29)(16,49)(17,50)(18,46)(19,47)(20,48)(21,43)(22,44)(23,45)(24,41)(25,42)(51,62)(52,63)(53,64)(54,65)(55,61)(66,74)(67,75)(68,71)(69,72)(70,73);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(75)!( 2, 5)( 3, 4)( 6, 7)( 8,10)(11,13)(14,15)(16,19)(17,18)(21,25)(22,24)(27,30)(28,29)(31,32)(33,35)(36,38)(39,40)(41,44)(42,43)(46,50)(47,49)(52,55)(53,54)(56,57)(58,60)(61,63)(64,65)(66,69)(67,68)(71,75)(72,74);
s1 := Sym(75)!( 2, 7)( 3,13)( 4,19)( 5,25)( 6,21)( 9,14)(10,20)(11,16)(12,22)(18,23)(26,51)(27,57)(28,63)(29,69)(30,75)(31,71)(32,52)(33,58)(34,64)(35,70)(36,66)(37,72)(38,53)(39,59)(40,65)(41,61)(42,67)(43,73)(44,54)(45,60)(46,56)(47,62)(48,68)(49,74)(50,55);
s2 := Sym(75)!( 1,37)( 2,38)( 3,39)( 4,40)( 5,36)( 6,31)( 7,32)( 8,33)( 9,34)(10,35)(11,30)(12,26)(13,27)(14,28)(15,29)(16,49)(17,50)(18,46)(19,47)(20,48)(21,43)(22,44)(23,45)(24,41)(25,42)(51,62)(52,63)(53,64)(54,65)(55,61)(66,74)(67,75)(68,71)(69,72)(70,73);
poly := sub<Sym(75)|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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1 >;
References : None.
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