Polytope of Type {2,6,9}

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

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

Permutation Representation (Magma) :

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

to this polytope