Polytope of Type {6,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18,6}*1944c
if this polytope has a name.
Group : SmallGroup(1944,2345)
Rank : 4
Schlafli Type : {6,18,6}
Number of vertices, edges, etc : 6, 81, 81, 9
Order of s₀s₁s₂s₃ : 18
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 : {2,18,6}*648b, {6,6,6}*648a
   9-fold quotients : {2,6,6}*216
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 := ( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,47)
(11,46)(12,48)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,38)(20,37)(21,39)
(22,44)(23,43)(24,45)(25,41)(26,40)(27,42)(56,57)(58,61)(59,63)(60,62)(64,74)
(65,73)(66,75)(67,80)(68,79)(69,81)(70,77)(71,76)(72,78);;
s2 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(19,20)
(22,23)(25,26)(28,64)(29,66)(30,65)(31,67)(32,69)(33,68)(34,70)(35,72)(36,71)
(37,55)(38,57)(39,56)(40,58)(41,60)(42,59)(43,61)(44,63)(45,62)(46,74)(47,73)
(48,75)(49,77)(50,76)(51,78)(52,80)(53,79)(54,81);;
s3 := ( 4, 8)( 5, 9)( 6, 7)(10,14)(11,15)(12,13)(19,27)(20,25)(21,26)(31,35)
(32,36)(33,34)(37,41)(38,42)(39,40)(46,54)(47,52)(48,53)(58,62)(59,63)(60,61)
(64,68)(65,69)(66,67)(73,81)(74,79)(75,80);;
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*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
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*s1*s2*s1*s2*s3*s2*s1*s3*s2*s3*s2 ];;
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)!( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)
(10,47)(11,46)(12,48)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,38)(20,37)
(21,39)(22,44)(23,43)(24,45)(25,41)(26,40)(27,42)(56,57)(58,61)(59,63)(60,62)
(64,74)(65,73)(66,75)(67,80)(68,79)(69,81)(70,77)(71,76)(72,78);
s2 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)
(19,20)(22,23)(25,26)(28,64)(29,66)(30,65)(31,67)(32,69)(33,68)(34,70)(35,72)
(36,71)(37,55)(38,57)(39,56)(40,58)(41,60)(42,59)(43,61)(44,63)(45,62)(46,74)
(47,73)(48,75)(49,77)(50,76)(51,78)(52,80)(53,79)(54,81);
s3 := Sym(81)!( 4, 8)( 5, 9)( 6, 7)(10,14)(11,15)(12,13)(19,27)(20,25)(21,26)
(31,35)(32,36)(33,34)(37,41)(38,42)(39,40)(46,54)(47,52)(48,53)(58,62)(59,63)
(60,61)(64,68)(65,69)(66,67)(73,81)(74,79)(75,80);
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*s0*s1*s0*s1*s2*s0*s1*s0*s1, 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*s1*s2*s1*s2*s3*s2*s1*s3*s2*s3*s2 >;

References : None.
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