Polytope of Type {6,9,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9,2}*1944a
if this polytope has a name.
Group : SmallGroup(1944,941)
Rank : 4
Schlafli Type : {6,9,2}
Number of vertices, edges, etc : 54, 243, 81, 2
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   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 : {6,9,2}*648b, {6,9,2}*648c
   9-fold quotients : {6,3,2}*216
   27-fold quotients : {6,3,2}*72
   81-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(10,22)(11,24)(12,23)(13,25)(14,27)(15,26)(16,19)
(17,21)(18,20)(28,55)(29,57)(30,56)(31,58)(32,60)(33,59)(34,61)(35,63)(36,62)
(37,76)(38,78)(39,77)(40,79)(41,81)(42,80)(43,73)(44,75)(45,74)(46,70)(47,72)
(48,71)(49,64)(50,66)(51,65)(52,67)(53,69)(54,68);;
s1 := ( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,39)
(11,38)(12,37)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,47)(20,46)(21,48)
(22,53)(23,52)(24,54)(25,50)(26,49)(27,51)(56,57)(58,61)(59,63)(60,62)(64,66)
(67,72)(68,71)(69,70)(73,74)(76,80)(77,79)(78,81);;
s2 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,16)(11,18)(12,17)(14,15)(19,22)(20,24)
(21,23)(26,27)(28,76)(29,78)(30,77)(31,73)(32,75)(33,74)(34,79)(35,81)(36,80)
(37,55)(38,57)(39,56)(40,61)(41,63)(42,62)(43,58)(44,60)(45,59)(46,70)(47,72)
(48,71)(49,67)(50,69)(51,68)(52,64)(53,66)(54,65);;
s3 := (82,83);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(83)!( 2, 3)( 5, 6)( 8, 9)(10,22)(11,24)(12,23)(13,25)(14,27)(15,26)
(16,19)(17,21)(18,20)(28,55)(29,57)(30,56)(31,58)(32,60)(33,59)(34,61)(35,63)
(36,62)(37,76)(38,78)(39,77)(40,79)(41,81)(42,80)(43,73)(44,75)(45,74)(46,70)
(47,72)(48,71)(49,64)(50,66)(51,65)(52,67)(53,69)(54,68);
s1 := Sym(83)!( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)
(10,39)(11,38)(12,37)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,47)(20,46)
(21,48)(22,53)(23,52)(24,54)(25,50)(26,49)(27,51)(56,57)(58,61)(59,63)(60,62)
(64,66)(67,72)(68,71)(69,70)(73,74)(76,80)(77,79)(78,81);
s2 := Sym(83)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,16)(11,18)(12,17)(14,15)(19,22)
(20,24)(21,23)(26,27)(28,76)(29,78)(30,77)(31,73)(32,75)(33,74)(34,79)(35,81)
(36,80)(37,55)(38,57)(39,56)(40,61)(41,63)(42,62)(43,58)(44,60)(45,59)(46,70)
(47,72)(48,71)(49,67)(50,69)(51,68)(52,64)(53,66)(54,65);
s3 := Sym(83)!(82,83);
poly := sub<Sym(83)|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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1 >;

to this polytope