Polytope of Type {6,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,6}*1296c
if this polytope has a name.
Group : SmallGroup(1296,2909)
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 : {2,12,6}*432a, {6,4,6}*432b
   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)(10,31)(11,33)(12,32)(13,28)(14,30)(15,29)
(16,34)(17,36)(18,35)(19,58)(20,60)(21,59)(22,55)(23,57)(24,56)(25,61)(26,63)
(27,62)(37,41)(38,40)(39,42)(43,44)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)
(52,72)(53,71)(54,70)(73,77)(74,76)(75,78)(79,80);;
s2 := ( 1, 2)( 4, 5)( 7, 8)(10,20)(11,19)(12,21)(13,23)(14,22)(15,24)(16,26)
(17,25)(18,27)(28,29)(31,32)(34,35)(37,47)(38,46)(39,48)(40,50)(41,49)(42,51)
(43,53)(44,52)(45,54)(55,56)(58,59)(61,62)(64,74)(65,73)(66,75)(67,77)(68,76)
(69,78)(70,80)(71,79)(72,81);;
s3 := ( 1,39)( 2,37)( 3,38)( 4,42)( 5,40)( 6,41)( 7,45)( 8,43)( 9,44)(10,29)
(11,30)(12,28)(13,32)(14,33)(15,31)(16,35)(17,36)(18,34)(19,46)(20,47)(21,48)
(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(55,66)(56,64)(57,65)(58,69)(59,67)
(60,68)(61,72)(62,70)(63,71);;
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, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*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, 
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)(10,31)(11,33)(12,32)(13,28)(14,30)
(15,29)(16,34)(17,36)(18,35)(19,58)(20,60)(21,59)(22,55)(23,57)(24,56)(25,61)
(26,63)(27,62)(37,41)(38,40)(39,42)(43,44)(46,69)(47,68)(48,67)(49,66)(50,65)
(51,64)(52,72)(53,71)(54,70)(73,77)(74,76)(75,78)(79,80);
s2 := Sym(81)!( 1, 2)( 4, 5)( 7, 8)(10,20)(11,19)(12,21)(13,23)(14,22)(15,24)
(16,26)(17,25)(18,27)(28,29)(31,32)(34,35)(37,47)(38,46)(39,48)(40,50)(41,49)
(42,51)(43,53)(44,52)(45,54)(55,56)(58,59)(61,62)(64,74)(65,73)(66,75)(67,77)
(68,76)(69,78)(70,80)(71,79)(72,81);
s3 := Sym(81)!( 1,39)( 2,37)( 3,38)( 4,42)( 5,40)( 6,41)( 7,45)( 8,43)( 9,44)
(10,29)(11,30)(12,28)(13,32)(14,33)(15,31)(16,35)(17,36)(18,34)(19,46)(20,47)
(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(55,66)(56,64)(57,65)(58,69)
(59,67)(60,68)(61,72)(62,70)(63,71);
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, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*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, 
s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3 >;

References : None.
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