Polytope of Type {6,9,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9,2}*1944c
if this polytope has a name.
Group : SmallGroup(1944,944)
Rank : 4
Schlafli Type : {6,9,2}
Number of vertices, edges, etc : 54, 243, 81, 2
Order of s₀s₁s₂s₃ : 18
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, {6,3,2}*648
   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 := ( 4, 7)( 5, 8)( 6, 9)(10,20)(11,21)(12,19)(13,26)(14,27)(15,25)(16,23)
(17,24)(18,22)(31,34)(32,35)(33,36)(37,47)(38,48)(39,46)(40,53)(41,54)(42,52)
(43,50)(44,51)(45,49)(58,61)(59,62)(60,63)(64,74)(65,75)(66,73)(67,80)(68,81)
(69,79)(70,77)(71,78)(72,76);;
s1 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,16)(11,18)(12,17)(14,15)(19,22)(20,24)
(21,23)(26,27)(28,77)(29,76)(30,78)(31,74)(32,73)(33,75)(34,80)(35,79)(36,81)
(37,56)(38,55)(39,57)(40,62)(41,61)(42,63)(43,59)(44,58)(45,60)(46,71)(47,70)
(48,72)(49,68)(50,67)(51,69)(52,65)(53,64)(54,66);;
s2 := ( 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)(55,56)(58,62)(59,61)(60,63)(65,66)
(67,70)(68,72)(69,71)(73,75)(76,81)(77,80)(78,79);;
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*s2*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope