Polytope of Type {6,6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,9}*1944d
if this polytope has a name.
Group : SmallGroup(1944,2340)
Rank : 4
Schlafli Type : {6,6,9}
Number of vertices, edges, etc : 18, 54, 81, 9
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 6
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 : {6,6,9}*648a, {6,6,9}*648b, {6,6,3}*648e
   6-fold quotients : {3,6,9}*324
   9-fold quotients : {2,6,9}*216, {6,2,9}*216, {6,6,3}*216a, {6,6,3}*216b
   18-fold quotients : {3,2,9}*108, {3,6,3}*108
   27-fold quotients : {2,2,9}*72, {2,6,3}*72, {6,2,3}*72
   54-fold quotients : {3,2,3}*36
   81-fold quotients : {2,2,3}*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 := ( 2, 3)( 5, 6)( 8, 9)(10,11)(13,14)(16,17)(19,21)(22,24)(25,27)(28,55)
(29,57)(30,56)(31,58)(32,60)(33,59)(34,61)(35,63)(36,62)(37,65)(38,64)(39,66)
(40,68)(41,67)(42,69)(43,71)(44,70)(45,72)(46,75)(47,74)(48,73)(49,78)(50,77)
(51,76)(52,81)(53,80)(54,79);;
s2 := ( 1,37)( 2,39)( 3,38)( 4,43)( 5,45)( 6,44)( 7,40)( 8,42)( 9,41)(10,28)
(11,30)(12,29)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,49)(20,51)(21,50)
(22,46)(23,48)(24,47)(25,52)(26,54)(27,53)(55,64)(56,66)(57,65)(58,70)(59,72)
(60,71)(61,67)(62,69)(63,68)(73,76)(74,78)(75,77)(80,81);;
s3 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)
(16,25)(17,27)(18,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)
(36,59)(37,76)(38,78)(39,77)(40,73)(41,75)(42,74)(43,79)(44,81)(45,80)(46,67)
(47,69)(48,68)(49,64)(50,66)(51,65)(52,70)(53,72)(54,71);;
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*s1*s2*s3*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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)!( 2, 3)( 5, 6)( 8, 9)(10,11)(13,14)(16,17)(19,21)(22,24)(25,27)
(28,55)(29,57)(30,56)(31,58)(32,60)(33,59)(34,61)(35,63)(36,62)(37,65)(38,64)
(39,66)(40,68)(41,67)(42,69)(43,71)(44,70)(45,72)(46,75)(47,74)(48,73)(49,78)
(50,77)(51,76)(52,81)(53,80)(54,79);
s2 := Sym(81)!( 1,37)( 2,39)( 3,38)( 4,43)( 5,45)( 6,44)( 7,40)( 8,42)( 9,41)
(10,28)(11,30)(12,29)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,49)(20,51)
(21,50)(22,46)(23,48)(24,47)(25,52)(26,54)(27,53)(55,64)(56,66)(57,65)(58,70)
(59,72)(60,71)(61,67)(62,69)(63,68)(73,76)(74,78)(75,77)(80,81);
s3 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,22)(11,24)(12,23)(13,19)(14,21)
(15,20)(16,25)(17,27)(18,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)
(35,60)(36,59)(37,76)(38,78)(39,77)(40,73)(41,75)(42,74)(43,79)(44,81)(45,80)
(46,67)(47,69)(48,68)(49,64)(50,66)(51,65)(52,70)(53,72)(54,71);
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*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope