Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1296s
if this polytope has a name.
Group : SmallGroup(1296,3528)
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
   Self-Petrie
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}*432h
   9-fold quotients : {12,6}*144c, {4,6}*144
   18-fold quotients : {4,6}*72, {6,6}*72b
   27-fold quotients : {4,6}*48a
   36-fold quotients : {6,3}*36
   54-fold quotients : {2,6}*24
   81-fold quotients : {4,2}*16
   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 := ( 4, 7)( 5, 8)( 6, 9)(10,28)(11,29)(12,30)(13,34)(14,35)(15,36)(16,31)
(17,32)(18,33)(19,55)(20,56)(21,57)(22,61)(23,62)(24,63)(25,58)(26,59)(27,60)
(40,43)(41,44)(42,45)(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)(53,68)
(54,69)(76,79)(77,80)(78,81);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)(20,24)
(21,23)(26,27)(28,58)(29,60)(30,59)(31,55)(32,57)(33,56)(34,61)(35,63)(36,62)
(37,67)(38,69)(39,68)(40,64)(41,66)(42,65)(43,70)(44,72)(45,71)(46,76)(47,78)
(48,77)(49,73)(50,75)(51,74)(52,79)(53,81)(54,80);;
s2 := ( 1,38)( 2,37)( 3,39)( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)(10,29)
(11,28)(12,30)(13,35)(14,34)(15,36)(16,32)(17,31)(18,33)(19,47)(20,46)(21,48)
(22,53)(23,52)(24,54)(25,50)(26,49)(27,51)(55,65)(56,64)(57,66)(58,71)(59,70)
(60,72)(61,68)(62,67)(63,69)(73,74)(76,80)(77,79)(78,81);;
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, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*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)!( 4, 7)( 5, 8)( 6, 9)(10,28)(11,29)(12,30)(13,34)(14,35)(15,36)
(16,31)(17,32)(18,33)(19,55)(20,56)(21,57)(22,61)(23,62)(24,63)(25,58)(26,59)
(27,60)(40,43)(41,44)(42,45)(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)
(53,68)(54,69)(76,79)(77,80)(78,81);
s1 := Sym(81)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)
(20,24)(21,23)(26,27)(28,58)(29,60)(30,59)(31,55)(32,57)(33,56)(34,61)(35,63)
(36,62)(37,67)(38,69)(39,68)(40,64)(41,66)(42,65)(43,70)(44,72)(45,71)(46,76)
(47,78)(48,77)(49,73)(50,75)(51,74)(52,79)(53,81)(54,80);
s2 := Sym(81)!( 1,38)( 2,37)( 3,39)( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)
(10,29)(11,28)(12,30)(13,35)(14,34)(15,36)(16,32)(17,31)(18,33)(19,47)(20,46)
(21,48)(22,53)(23,52)(24,54)(25,50)(26,49)(27,51)(55,65)(56,64)(57,66)(58,71)
(59,70)(60,72)(61,68)(62,67)(63,69)(73,74)(76,80)(77,79)(78,81);
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, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*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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