Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1296t
if this polytope has a name.
Group : SmallGroup(1296,3529)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 108, 324, 54
Order of s₀s₁s₂ : 12
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 : {4,6}*432b, {12,6}*432i
   6-fold quotients : {12,6}*216c
   9-fold quotients : {12,6}*144a, {4,6}*144
   18-fold quotients : {4,6}*72, {6,6}*72a
   27-fold quotients : {12,2}*48, {4,6}*48a
   54-fold quotients : {2,6}*24, {6,2}*24
   81-fold quotients : {4,2}*16
   108-fold quotients : {2,3}*12, {3,2}*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)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(28,55)
(29,57)(30,56)(31,58)(32,60)(33,59)(34,61)(35,63)(36,62)(37,64)(38,66)(39,65)
(40,67)(41,69)(42,68)(43,70)(44,72)(45,71)(46,73)(47,75)(48,74)(49,76)(50,78)
(51,77)(52,79)(53,81)(54,80);;
s1 := ( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,29)(11,28)(12,30)(13,35)(14,34)(15,36)
(16,32)(17,31)(18,33)(19,56)(20,55)(21,57)(22,62)(23,61)(24,63)(25,59)(26,58)
(27,60)(37,38)(40,44)(41,43)(42,45)(46,65)(47,64)(48,66)(49,71)(50,70)(51,72)
(52,68)(53,67)(54,69)(73,74)(76,80)(77,79)(78,81);;
s2 := ( 1,13)( 2,14)( 3,15)( 4,10)( 5,11)( 6,12)( 7,16)( 8,17)( 9,18)(19,22)
(20,23)(21,24)(28,67)(29,68)(30,69)(31,64)(32,65)(33,66)(34,70)(35,71)(36,72)
(37,58)(38,59)(39,60)(40,55)(41,56)(42,57)(43,61)(44,62)(45,63)(46,76)(47,77)
(48,78)(49,73)(50,74)(51,75)(52,79)(53,80)(54,81);;
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*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
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