Polytope of Type {9,6,6}

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

References : None.
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