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}*1800b
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}*600a
   6-fold quotients : {6,3}*300
   25-fold quotients : {6,6}*72c
   50-fold quotients : {3,6}*36
   75-fold quotients : {6,2}*24
   150-fold quotients : {3,2}*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=<s0*s1*s0*s1*s0*s1> of order 2.
      78 facets:
         6 of {3}*6
         72 of {6}*12
      75 vertex figures:
         75 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 2.
      75 facets:
         75 of {6}*12
      100 vertex figures:
         50 of {6}*12
         50 of {3}*6
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 3.
      50 facets:
         50 of {6}*12
      54 vertex figures:
         48 of {6}*12
         6 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 5.
      30 facets:
         30 of {6}*12
      30 vertex figures:
         30 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 5.
      30 facets:
         30 of {6}*12
      30 vertex figures:
         30 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1, s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 10.
      18 facets:
         6 of {3}*6
         12 of {6}*12
      15 vertex figures:
         15 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 10.
      15 facets:
         15 of {6}*12
      20 vertex figures:
         10 of {6}*12
         10 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 10.
      18 facets:
         6 of {3}*6
         12 of {6}*12
      15 vertex figures:
         15 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)(26,51)(27,55)(28,54)(29,53)(30,52)(31,57)(32,56)(33,60)(34,59)(35,58)(36,63)(37,62)(38,61)(39,65)(40,64)(41,69)(42,68)(43,67)(44,66)(45,70)(46,75)(47,74)(48,73)(49,72)(50,71);;
s1 := ( 1,26)( 2,50)( 3,44)( 4,38)( 5,32)( 6,31)( 7,30)( 8,49)( 9,43)(10,37)(11,36)(12,35)(13,29)(14,48)(15,42)(16,41)(17,40)(18,34)(19,28)(20,47)(21,46)(22,45)(23,39)(24,33)(25,27)(52,75)(53,69)(54,63)(55,57)(58,74)(59,68)(60,62)(64,73)(65,67)(70,72);;
s2 := ( 1,12)( 2,13)( 3,14)( 4,15)( 5,11)(16,24)(17,25)(18,21)(19,22)(20,23)(26,37)(27,38)(28,39)(29,40)(30,36)(41,49)(42,50)(43,46)(44,47)(45,48)(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, 
s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*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)(26,51)(27,55)(28,54)(29,53)(30,52)(31,57)(32,56)(33,60)(34,59)(35,58)(36,63)(37,62)(38,61)(39,65)(40,64)(41,69)(42,68)(43,67)(44,66)(45,70)(46,75)(47,74)(48,73)(49,72)(50,71);
s1 := Sym(75)!( 1,26)( 2,50)( 3,44)( 4,38)( 5,32)( 6,31)( 7,30)( 8,49)( 9,43)(10,37)(11,36)(12,35)(13,29)(14,48)(15,42)(16,41)(17,40)(18,34)(19,28)(20,47)(21,46)(22,45)(23,39)(24,33)(25,27)(52,75)(53,69)(54,63)(55,57)(58,74)(59,68)(60,62)(64,73)(65,67)(70,72);
s2 := Sym(75)!( 1,12)( 2,13)( 3,14)( 4,15)( 5,11)(16,24)(17,25)(18,21)(19,22)(20,23)(26,37)(27,38)(28,39)(29,40)(30,36)(41,49)(42,50)(43,46)(44,47)(45,48)(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, 
s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1 >;
References : None.
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