Polytope of Type {6,9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9,6}*1944d
if this polytope has a name.
Group : SmallGroup(1944,2344)
Rank : 4
Schlafli Type : {6,9,6}
Number of vertices, edges, etc : 6, 81, 81, 18
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 : {2,9,6}*648b, {6,3,6}*648a
   9-fold quotients : {2,3,6}*216, {6,3,6}*216
   27-fold quotients : {2,3,6}*72, {6,3,2}*72
   81-fold quotients : {2,3,2}*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)(10,46)
(11,48)(12,47)(13,49)(14,51)(15,50)(16,52)(17,54)(18,53)(19,37)(20,39)(21,38)
(22,40)(23,42)(24,41)(25,43)(26,45)(27,44)(56,57)(59,60)(62,63)(64,73)(65,75)
(66,74)(67,76)(68,78)(69,77)(70,79)(71,81)(72,80);;
s2 := ( 1,13)( 2,15)( 3,14)( 4,17)( 5,16)( 6,18)( 7,12)( 8,11)( 9,10)(19,20)
(22,24)(26,27)(28,67)(29,69)(30,68)(31,71)(32,70)(33,72)(34,66)(35,65)(36,64)
(37,63)(38,62)(39,61)(40,55)(41,57)(42,56)(43,59)(44,58)(45,60)(46,74)(47,73)
(48,75)(49,78)(50,77)(51,76)(52,79)(53,81)(54,80);;
s3 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)
(16,22)(17,24)(18,23)(29,30)(31,34)(32,36)(33,35)(37,46)(38,48)(39,47)(40,52)
(41,54)(42,53)(43,49)(44,51)(45,50)(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);;
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*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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)( 4,31)( 5,33)( 6,32)( 7,34)( 8,36)( 9,35)
(10,46)(11,48)(12,47)(13,49)(14,51)(15,50)(16,52)(17,54)(18,53)(19,37)(20,39)
(21,38)(22,40)(23,42)(24,41)(25,43)(26,45)(27,44)(56,57)(59,60)(62,63)(64,73)
(65,75)(66,74)(67,76)(68,78)(69,77)(70,79)(71,81)(72,80);
s2 := Sym(81)!( 1,13)( 2,15)( 3,14)( 4,17)( 5,16)( 6,18)( 7,12)( 8,11)( 9,10)
(19,20)(22,24)(26,27)(28,67)(29,69)(30,68)(31,71)(32,70)(33,72)(34,66)(35,65)
(36,64)(37,63)(38,62)(39,61)(40,55)(41,57)(42,56)(43,59)(44,58)(45,60)(46,74)
(47,73)(48,75)(49,78)(50,77)(51,76)(52,79)(53,81)(54,80);
s3 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)
(15,26)(16,22)(17,24)(18,23)(29,30)(31,34)(32,36)(33,35)(37,46)(38,48)(39,47)
(40,52)(41,54)(42,53)(43,49)(44,51)(45,50)(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);
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*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope