Polytope of Type {12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1296b
if this polytope has a name.
Group : SmallGroup(1296,2909)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 54, 324, 54
Order of s₀s₁s₂ : 6
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 : {4,12}*432b
   9-fold quotients : {4,4}*144
   18-fold quotients : {4,4}*72
   54-fold quotients : {2,6}*24
   108-fold quotients : {2,3}*12
   162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)
(17,36)(18,35)(19,55)(20,57)(21,56)(22,58)(23,60)(24,59)(25,61)(26,63)(27,62)
(37,38)(40,41)(43,44)(46,66)(47,65)(48,64)(49,69)(50,68)(51,67)(52,72)(53,71)
(54,70)(73,74)(76,77)(79,80);;
s1 := ( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,38)
(11,37)(12,39)(13,44)(14,43)(15,45)(16,41)(17,40)(18,42)(19,48)(20,47)(21,46)
(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(56,57)(58,61)(59,63)(60,62)(64,65)
(67,71)(68,70)(69,72)(73,75)(76,81)(77,80)(78,79);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,58)(11,60)(12,59)(13,55)(14,57)(15,56)
(16,61)(17,63)(18,62)(19,31)(20,33)(21,32)(22,28)(23,30)(24,29)(25,34)(26,36)
(27,35)(37,77)(38,76)(39,78)(40,74)(41,73)(42,75)(43,80)(44,79)(45,81)(46,51)
(47,50)(48,49)(52,54)(64,69)(65,68)(66,67)(70,72);;
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, s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(81)!( 2, 3)( 5, 6)( 8, 9)(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)
(16,34)(17,36)(18,35)(19,55)(20,57)(21,56)(22,58)(23,60)(24,59)(25,61)(26,63)
(27,62)(37,38)(40,41)(43,44)(46,66)(47,65)(48,64)(49,69)(50,68)(51,67)(52,72)
(53,71)(54,70)(73,74)(76,77)(79,80);
s1 := Sym(81)!( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)
(10,38)(11,37)(12,39)(13,44)(14,43)(15,45)(16,41)(17,40)(18,42)(19,48)(20,47)
(21,46)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(56,57)(58,61)(59,63)(60,62)
(64,65)(67,71)(68,70)(69,72)(73,75)(76,81)(77,80)(78,79);
s2 := Sym(81)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,58)(11,60)(12,59)(13,55)(14,57)
(15,56)(16,61)(17,63)(18,62)(19,31)(20,33)(21,32)(22,28)(23,30)(24,29)(25,34)
(26,36)(27,35)(37,77)(38,76)(39,78)(40,74)(41,73)(42,75)(43,80)(44,79)(45,81)
(46,51)(47,50)(48,49)(52,54)(64,69)(65,68)(66,67)(70,72);
poly := sub<Sym(81)|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, s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
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