Polytope of Type {6,9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9,6}*1944a
if this polytope has a name.
Group : SmallGroup(1944,2340)
Rank : 4
Schlafli Type : {6,9,6}
Number of vertices, edges, etc : 6, 81, 81, 18
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   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}*648a, {6,9,6}*648, {6,3,6}*648a
   9-fold quotients : {2,9,6}*216, {6,9,2}*216, {2,3,6}*216, {6,3,6}*216
   27-fold quotients : {2,9,2}*72, {2,3,6}*72, {6,3,2}*72
   81-fold quotients : {2,3,2}*24
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)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,49)
(11,50)(12,51)(13,46)(14,47)(15,48)(16,52)(17,53)(18,54)(19,40)(20,41)(21,42)
(22,37)(23,38)(24,39)(25,43)(26,44)(27,45)(58,61)(59,62)(60,63)(64,76)(65,77)
(66,78)(67,73)(68,74)(69,75)(70,79)(71,80)(72,81);;
s2 := ( 1,11)( 2,12)( 3,10)( 4,17)( 5,18)( 6,16)( 7,14)( 8,15)( 9,13)(19,22)
(20,23)(21,24)(28,65)(29,66)(30,64)(31,71)(32,72)(33,70)(34,68)(35,69)(36,67)
(37,57)(38,55)(39,56)(40,63)(41,61)(42,62)(43,60)(44,58)(45,59)(46,76)(47,77)
(48,78)(49,73)(50,74)(51,75)(52,79)(53,80)(54,81);;
s3 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)
(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)(62,63)
(65,66)(68,69)(71,72)(74,75)(77,78)(80,81);;
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, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)
(10,49)(11,50)(12,51)(13,46)(14,47)(15,48)(16,52)(17,53)(18,54)(19,40)(20,41)
(21,42)(22,37)(23,38)(24,39)(25,43)(26,44)(27,45)(58,61)(59,62)(60,63)(64,76)
(65,77)(66,78)(67,73)(68,74)(69,75)(70,79)(71,80)(72,81);
s2 := Sym(81)!( 1,11)( 2,12)( 3,10)( 4,17)( 5,18)( 6,16)( 7,14)( 8,15)( 9,13)
(19,22)(20,23)(21,24)(28,65)(29,66)(30,64)(31,71)(32,72)(33,70)(34,68)(35,69)
(36,67)(37,57)(38,55)(39,56)(40,63)(41,61)(42,62)(43,60)(44,58)(45,59)(46,76)
(47,77)(48,78)(49,73)(50,74)(51,75)(52,79)(53,80)(54,81);
s3 := Sym(81)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)
(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)
(62,63)(65,66)(68,69)(71,72)(74,75)(77,78)(80,81);
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, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope