Polytope of Type {6,9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9,6}*1944b
if this polytope has a name.
Group : SmallGroup(1944,2340)
Rank : 4
Schlafli Type : {6,9,6}
Number of vertices, edges, etc : 18, 81, 81, 6
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 : {6,9,2}*648a, {6,9,6}*648, {6,3,6}*648b
   9-fold quotients : {2,9,6}*216, {6,9,2}*216, {6,3,2}*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 := ( 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);;
s1 := ( 4, 7)( 5, 8)( 6, 9)(10,24)(11,22)(12,23)(13,21)(14,19)(15,20)(16,27)
(17,25)(18,26)(28,55)(29,56)(30,57)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)
(37,78)(38,76)(39,77)(40,75)(41,73)(42,74)(43,81)(44,79)(45,80)(46,68)(47,69)
(48,67)(49,65)(50,66)(51,64)(52,71)(53,72)(54,70);;
s2 := ( 1,37)( 2,38)( 3,39)( 4,43)( 5,44)( 6,45)( 7,40)( 8,41)( 9,42)(10,28)
(11,29)(12,30)(13,34)(14,35)(15,36)(16,31)(17,32)(18,33)(19,49)(20,50)(21,51)
(22,46)(23,47)(24,48)(25,52)(26,53)(27,54)(55,64)(56,65)(57,66)(58,70)(59,71)
(60,72)(61,67)(62,68)(63,69)(73,76)(74,77)(75,78);;
s3 := (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);;
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, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1, 
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)!( 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);
s1 := Sym(81)!( 4, 7)( 5, 8)( 6, 9)(10,24)(11,22)(12,23)(13,21)(14,19)(15,20)
(16,27)(17,25)(18,26)(28,55)(29,56)(30,57)(31,61)(32,62)(33,63)(34,58)(35,59)
(36,60)(37,78)(38,76)(39,77)(40,75)(41,73)(42,74)(43,81)(44,79)(45,80)(46,68)
(47,69)(48,67)(49,65)(50,66)(51,64)(52,71)(53,72)(54,70);
s2 := Sym(81)!( 1,37)( 2,38)( 3,39)( 4,43)( 5,44)( 6,45)( 7,40)( 8,41)( 9,42)
(10,28)(11,29)(12,30)(13,34)(14,35)(15,36)(16,31)(17,32)(18,33)(19,49)(20,50)
(21,51)(22,46)(23,47)(24,48)(25,52)(26,53)(27,54)(55,64)(56,65)(57,66)(58,70)
(59,71)(60,72)(61,67)(62,68)(63,69)(73,76)(74,77)(75,78);
s3 := 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);
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, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope