Polytope of Type {6,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,4}*1296d
if this polytope has a name.
Group : SmallGroup(1296,3531)
Rank : 4
Schlafli Type : {6,6,4}
Number of vertices, edges, etc : 18, 81, 54, 6
Order of s₀s₁s₂s₃ : 4
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   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) :
   9-fold quotients : {2,6,4}*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)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,46)
(11,48)(12,47)(13,52)(14,54)(15,53)(16,49)(17,51)(18,50)(19,37)(20,39)(21,38)
(22,43)(23,45)(24,44)(25,40)(26,42)(27,41)(56,57)(58,61)(59,63)(60,62)(64,73)
(65,75)(66,74)(67,79)(68,81)(69,80)(70,76)(71,78)(72,77);;
s2 := ( 1, 2)( 4, 6)( 8, 9)(10,20)(11,19)(12,21)(13,24)(14,23)(15,22)(16,25)
(17,27)(18,26)(28,38)(29,37)(30,39)(31,42)(32,41)(33,40)(34,43)(35,45)(36,44)
(46,47)(49,51)(53,54)(55,74)(56,73)(57,75)(58,78)(59,77)(60,76)(61,79)(62,81)
(63,80)(64,65)(67,69)(71,72);;
s3 := ( 2, 8)( 3, 6)( 4, 7)(10,46)(11,53)(12,51)(13,52)(14,50)(15,48)(16,49)
(17,47)(18,54)(19,64)(20,71)(21,69)(22,70)(23,68)(24,66)(25,67)(26,65)(27,72)
(29,35)(30,33)(31,34)(37,73)(38,80)(39,78)(40,79)(41,77)(42,75)(43,76)(44,74)
(45,81)(56,62)(57,60)(58,61);;
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*s2*s3, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s2*s0*s3*s2*s1*s0*s1*s2*s3*s2*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*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)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)
(10,46)(11,48)(12,47)(13,52)(14,54)(15,53)(16,49)(17,51)(18,50)(19,37)(20,39)
(21,38)(22,43)(23,45)(24,44)(25,40)(26,42)(27,41)(56,57)(58,61)(59,63)(60,62)
(64,73)(65,75)(66,74)(67,79)(68,81)(69,80)(70,76)(71,78)(72,77);
s2 := Sym(81)!( 1, 2)( 4, 6)( 8, 9)(10,20)(11,19)(12,21)(13,24)(14,23)(15,22)
(16,25)(17,27)(18,26)(28,38)(29,37)(30,39)(31,42)(32,41)(33,40)(34,43)(35,45)
(36,44)(46,47)(49,51)(53,54)(55,74)(56,73)(57,75)(58,78)(59,77)(60,76)(61,79)
(62,81)(63,80)(64,65)(67,69)(71,72);
s3 := Sym(81)!( 2, 8)( 3, 6)( 4, 7)(10,46)(11,53)(12,51)(13,52)(14,50)(15,48)
(16,49)(17,47)(18,54)(19,64)(20,71)(21,69)(22,70)(23,68)(24,66)(25,67)(26,65)
(27,72)(29,35)(30,33)(31,34)(37,73)(38,80)(39,78)(40,79)(41,77)(42,75)(43,76)
(44,74)(45,81)(56,62)(57,60)(58,61);
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*s2*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s2*s0*s3*s2*s1*s0*s1*s2*s3*s2*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 >;

References : None.
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