Polytope of Type {6,6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,9}*1944g
if this polytope has a name.
Group : SmallGroup(1944,2344)
Rank : 4
Schlafli Type : {6,6,9}
Number of vertices, edges, etc : 6, 54, 81, 27
Order of s₀s₁s₂s₃ : 6
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}*648b, {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)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,46)
(11,48)(12,47)(13,52)(14,54)(15,53)(16,49)(17,51)(18,50)(19,37)(20,39)(21,38)
(22,43)(23,45)(24,44)(25,40)(26,42)(27,41)(56,57)(58,61)(59,63)(60,62)(64,73)
(65,75)(66,74)(67,79)(68,81)(69,80)(70,76)(71,78)(72,77);;
s2 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)
(23,24)(26,27)(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(34,43)(35,45)(36,44)
(47,48)(50,51)(53,54)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)(62,72)
(63,71)(74,75)(77,78)(80,81);;
s3 := ( 2, 3)( 4, 6)( 7, 8)(10,27)(11,26)(12,25)(13,20)(14,19)(15,21)(16,22)
(17,24)(18,23)(29,30)(31,33)(34,35)(37,54)(38,53)(39,52)(40,47)(41,46)(42,48)
(43,49)(44,51)(45,50)(56,57)(58,60)(61,62)(64,81)(65,80)(66,79)(67,74)(68,73)
(69,75)(70,76)(71,78)(72,77);;
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, 
s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*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)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)
(10,46)(11,48)(12,47)(13,52)(14,54)(15,53)(16,49)(17,51)(18,50)(19,37)(20,39)
(21,38)(22,43)(23,45)(24,44)(25,40)(26,42)(27,41)(56,57)(58,61)(59,63)(60,62)
(64,73)(65,75)(66,74)(67,79)(68,81)(69,80)(70,76)(71,78)(72,77);
s2 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)
(20,21)(23,24)(26,27)(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(34,43)(35,45)
(36,44)(47,48)(50,51)(53,54)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)
(62,72)(63,71)(74,75)(77,78)(80,81);
s3 := Sym(81)!( 2, 3)( 4, 6)( 7, 8)(10,27)(11,26)(12,25)(13,20)(14,19)(15,21)
(16,22)(17,24)(18,23)(29,30)(31,33)(34,35)(37,54)(38,53)(39,52)(40,47)(41,46)
(42,48)(43,49)(44,51)(45,50)(56,57)(58,60)(61,62)(64,81)(65,80)(66,79)(67,74)
(68,73)(69,75)(70,76)(71,78)(72,77);
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, 
s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2 >;

References : None.
to this polytope