Polytope of Type {9,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,6}*1944e
if this polytope has a name.
Group : SmallGroup(1944,2341)
Rank : 4
Schlafli Type : {9,6,6}
Number of vertices, edges, etc : 9, 81, 54, 18
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   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 : {9,6,6}*648b, {3,6,6}*648d
   9-fold quotients : {9,2,6}*216, {9,6,2}*216, {3,6,6}*216b
   18-fold quotients : {9,2,3}*108
   27-fold quotients : {9,2,2}*72, {3,2,6}*72, {3,6,2}*72
   54-fold quotients : {3,2,3}*36
   81-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope