Polytope of Type {6,6,9}

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

References : None.
to this polytope