Polytope of Type {6,6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,9}*1944h
if this polytope has a name.
Group : SmallGroup(1944,2345)
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}*648d, {6,6,3}*648c
   9-fold quotients : {2,6,3}*216, {6,6,3}*216b
   27-fold quotients : {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)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,37)
(11,38)(12,39)(13,43)(14,44)(15,45)(16,40)(17,41)(18,42)(19,46)(20,47)(21,48)
(22,52)(23,53)(24,54)(25,49)(26,50)(27,51)(58,61)(59,62)(60,63)(67,70)(68,71)
(69,72)(76,79)(77,80)(78,81);;
s2 := ( 2, 3)( 4, 6)( 7, 8)(10,25)(11,27)(12,26)(13,21)(14,20)(15,19)(16,23)
(17,22)(18,24)(29,30)(31,33)(34,35)(37,52)(38,54)(39,53)(40,48)(41,47)(42,46)
(43,50)(44,49)(45,51)(56,57)(58,60)(61,62)(64,79)(65,81)(66,80)(67,75)(68,74)
(69,73)(70,77)(71,76)(72,78);;
s3 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(19,20)
(22,23)(25,26)(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(34,43)(35,45)(36,44)
(46,47)(49,50)(52,53)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)(62,72)
(63,71)(73,74)(76,77)(79,80);;
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, 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, 
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*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)
(10,37)(11,38)(12,39)(13,43)(14,44)(15,45)(16,40)(17,41)(18,42)(19,46)(20,47)
(21,48)(22,52)(23,53)(24,54)(25,49)(26,50)(27,51)(58,61)(59,62)(60,63)(67,70)
(68,71)(69,72)(76,79)(77,80)(78,81);
s2 := Sym(81)!( 2, 3)( 4, 6)( 7, 8)(10,25)(11,27)(12,26)(13,21)(14,20)(15,19)
(16,23)(17,22)(18,24)(29,30)(31,33)(34,35)(37,52)(38,54)(39,53)(40,48)(41,47)
(42,46)(43,50)(44,49)(45,51)(56,57)(58,60)(61,62)(64,79)(65,81)(66,80)(67,75)
(68,74)(69,73)(70,77)(71,76)(72,78);
s3 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)
(19,20)(22,23)(25,26)(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(34,43)(35,45)
(36,44)(46,47)(49,50)(52,53)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)
(62,72)(63,71)(73,74)(76,77)(79,80);
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, 
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, 
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;

References : None.
to this polytope