Polytope of Type {4,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,3}*1296b
if this polytope has a name.
Group : SmallGroup(1296,3528)
Rank : 4
Schlafli Type : {4,6,3}
Number of vertices, edges, etc : 36, 108, 81, 3
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   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 : {4,6,3}*144
   18-fold quotients : {2,6,3}*72
   27-fold quotients : {4,2,3}*48
   54-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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