Polytope of Type {6,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,6}*1296a
if this polytope has a name.
Group : SmallGroup(1296,2909)
Rank : 4
Schlafli Type : {6,4,6}
Number of vertices, edges, etc : 27, 54, 54, 6
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 : {6,4,2}*432, {6,4,6}*432a
   9-fold quotients : {6,4,2}*144
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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