Polytope of Type {6,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,6}*1296i
if this polytope has a name.
Group : SmallGroup(1296,3529)
Rank : 4
Schlafli Type : {6,12,6}
Number of vertices, edges, etc : 6, 54, 54, 9
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,4,6}*432b, {2,12,6}*432c
   9-fold quotients : {2,4,6}*144
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)(60,63)
(67,70)(68,71)(69,72)(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, 2)( 4, 5)( 7, 8)(10,29)(11,28)(12,30)(13,32)(14,31)(15,33)(16,35)
(17,34)(18,36)(19,56)(20,55)(21,57)(22,59)(23,58)(24,60)(25,62)(26,61)(27,63)
(37,38)(40,41)(43,44)(46,65)(47,64)(48,66)(49,68)(50,67)(51,69)(52,71)(53,70)
(54,72)(73,74)(76,77)(79,80);;
s3 := ( 1,10)( 2,11)( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18)(28,64)
(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,55)(38,56)(39,57)
(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,73)(47,74)(48,75)(49,76)(50,77)
(51,78)(52,79)(53,80)(54,81);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(81)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)
(60,63)(67,70)(68,71)(69,72)(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, 2)( 4, 5)( 7, 8)(10,29)(11,28)(12,30)(13,32)(14,31)(15,33)
(16,35)(17,34)(18,36)(19,56)(20,55)(21,57)(22,59)(23,58)(24,60)(25,62)(26,61)
(27,63)(37,38)(40,41)(43,44)(46,65)(47,64)(48,66)(49,68)(50,67)(51,69)(52,71)
(53,70)(54,72)(73,74)(76,77)(79,80);
s3 := Sym(81)!( 1,10)( 2,11)( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18)
(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,55)(38,56)
(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,73)(47,74)(48,75)(49,76)
(50,77)(51,78)(52,79)(53,80)(54,81);
poly := sub<Sym(81)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3 >;

References : None.
to this polytope