Polytope of Type {2,9,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,6,6}*1296b
if this polytope has a name.
Group : SmallGroup(1296,2984)
Rank : 5
Schlafli Type : {2,9,6,6}
Number of vertices, edges, etc : 2, 9, 27, 18, 6
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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 : {2,9,2,6}*432, {2,9,6,2}*432, {2,3,6,6}*432b
   6-fold quotients : {2,9,2,3}*216
   9-fold quotients : {2,9,2,2}*144, {2,3,2,6}*144, {2,3,6,2}*144
   18-fold quotients : {2,3,2,3}*72
   27-fold quotients : {2,3,2,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 4, 5)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19)(22,23)(24,27)
(25,29)(26,28)(30,59)(31,58)(32,57)(33,65)(34,64)(35,63)(36,62)(37,61)(38,60)
(39,68)(40,67)(41,66)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,77)(49,76)
(50,75)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78);;
s2 := ( 3,33)( 4,35)( 5,34)( 6,30)( 7,32)( 8,31)( 9,36)(10,38)(11,37)(12,42)
(13,44)(14,43)(15,39)(16,41)(17,40)(18,45)(19,47)(20,46)(21,51)(22,53)(23,52)
(24,48)(25,50)(26,49)(27,54)(28,56)(29,55)(57,62)(58,61)(59,60)(63,65)(66,71)
(67,70)(68,69)(72,74)(75,80)(76,79)(77,78)(81,83);;
s3 := ( 6, 9)( 7,10)( 8,11)(12,21)(13,22)(14,23)(15,27)(16,28)(17,29)(18,24)
(19,25)(20,26)(33,36)(34,37)(35,38)(39,48)(40,49)(41,50)(42,54)(43,55)(44,56)
(45,51)(46,52)(47,53)(60,63)(61,64)(62,65)(66,75)(67,76)(68,77)(69,81)(70,82)
(71,83)(72,78)(73,79)(74,80);;
s4 := ( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18)(10,19)(11,20)(30,39)
(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(57,66)(58,67)(59,68)
(60,69)(61,70)(62,71)(63,72)(64,73)(65,74);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
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(83)!(1,2);
s1 := Sym(83)!( 4, 5)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19)(22,23)
(24,27)(25,29)(26,28)(30,59)(31,58)(32,57)(33,65)(34,64)(35,63)(36,62)(37,61)
(38,60)(39,68)(40,67)(41,66)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,77)
(49,76)(50,75)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78);
s2 := Sym(83)!( 3,33)( 4,35)( 5,34)( 6,30)( 7,32)( 8,31)( 9,36)(10,38)(11,37)
(12,42)(13,44)(14,43)(15,39)(16,41)(17,40)(18,45)(19,47)(20,46)(21,51)(22,53)
(23,52)(24,48)(25,50)(26,49)(27,54)(28,56)(29,55)(57,62)(58,61)(59,60)(63,65)
(66,71)(67,70)(68,69)(72,74)(75,80)(76,79)(77,78)(81,83);
s3 := Sym(83)!( 6, 9)( 7,10)( 8,11)(12,21)(13,22)(14,23)(15,27)(16,28)(17,29)
(18,24)(19,25)(20,26)(33,36)(34,37)(35,38)(39,48)(40,49)(41,50)(42,54)(43,55)
(44,56)(45,51)(46,52)(47,53)(60,63)(61,64)(62,65)(66,75)(67,76)(68,77)(69,81)
(70,82)(71,83)(72,78)(73,79)(74,80);
s4 := Sym(83)!( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18)(10,19)(11,20)
(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(57,66)(58,67)
(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(65,74);
poly := sub<Sym(83)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope