Polytope of Type {12,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,3}*1296f
if this polytope has a name.
Group : SmallGroup(1296,3529)
Rank : 4
Schlafli Type : {12,6,3}
Number of vertices, edges, etc : 36, 108, 27, 3
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   Universal
   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 : {4,6,3}*432b
   9-fold quotients : {12,2,3}*144
   18-fold quotients : {6,2,3}*72
   27-fold quotients : {4,2,3}*48
   36-fold quotients : {3,2,3}*36
   54-fold quotients : {2,2,3}*24
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, 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);;
s2 := ( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)(22,25)
(23,26)(24,27)(28,64)(29,65)(30,66)(31,70)(32,71)(33,72)(34,67)(35,68)(36,69)
(37,55)(38,56)(39,57)(40,61)(41,62)(42,63)(43,58)(44,59)(45,60)(46,73)(47,74)
(48,75)(49,79)(50,80)(51,81)(52,76)(53,77)(54,78);;
s3 := ( 1, 4)( 2, 5)( 3, 6)(10,22)(11,23)(12,24)(13,19)(14,20)(15,21)(16,25)
(17,26)(18,27)(28,58)(29,59)(30,60)(31,55)(32,56)(33,57)(34,61)(35,62)(36,63)
(37,76)(38,77)(39,78)(40,73)(41,74)(42,75)(43,79)(44,80)(45,81)(46,67)(47,68)
(48,69)(49,64)(50,65)(51,66)(52,70)(53,71)(54,72);;
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, s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s2*s1*s2*s0*s1*s2*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*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, 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);
s2 := Sym(81)!( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)
(22,25)(23,26)(24,27)(28,64)(29,65)(30,66)(31,70)(32,71)(33,72)(34,67)(35,68)
(36,69)(37,55)(38,56)(39,57)(40,61)(41,62)(42,63)(43,58)(44,59)(45,60)(46,73)
(47,74)(48,75)(49,79)(50,80)(51,81)(52,76)(53,77)(54,78);
s3 := Sym(81)!( 1, 4)( 2, 5)( 3, 6)(10,22)(11,23)(12,24)(13,19)(14,20)(15,21)
(16,25)(17,26)(18,27)(28,58)(29,59)(30,60)(31,55)(32,56)(33,57)(34,61)(35,62)
(36,63)(37,76)(38,77)(39,78)(40,73)(41,74)(42,75)(43,79)(44,80)(45,81)(46,67)
(47,68)(48,69)(49,64)(50,65)(51,66)(52,70)(53,71)(54,72);
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, 
s2*s3*s2*s3*s2*s3, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s2*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*s0*s1 >;

References : None.
to this polytope