Polytope of Type {2,9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,6}*1944b
if this polytope has a name.
Group : SmallGroup(1944,943)
Rank : 4
Schlafli Type : {2,9,6}
Number of vertices, edges, etc : 2, 81, 243, 54
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 : {2,9,6}*648b, {2,9,6}*648d
   9-fold quotients : {2,3,6}*216
   27-fold quotients : {2,3,6}*72
   81-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope