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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,9,6}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1858)
Rank : 5
Schlafli Type : {2,2,9,6}
Number of vertices, edges, etc : 2, 2, 27, 81, 18
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 : {2,2,9,6}*432, {2,2,3,6}*432
   9-fold quotients : {2,2,9,2}*144, {2,2,3,6}*144
   27-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (3,4);;
s2 := ( 8,11)( 9,12)(10,13)(14,23)(15,24)(16,25)(17,29)(18,30)(19,31)(20,26)
(21,27)(22,28)(32,62)(33,63)(34,64)(35,59)(36,60)(37,61)(38,65)(39,66)(40,67)
(41,80)(42,81)(43,82)(44,77)(45,78)(46,79)(47,83)(48,84)(49,85)(50,71)(51,72)
(52,73)(53,68)(54,69)(55,70)(56,74)(57,75)(58,76);;
s3 := ( 5,43)( 6,41)( 7,42)( 8,49)( 9,47)(10,48)(11,46)(12,44)(13,45)(14,33)
(15,34)(16,32)(17,39)(18,40)(19,38)(20,36)(21,37)(22,35)(23,50)(24,51)(25,52)
(26,56)(27,57)(28,58)(29,53)(30,54)(31,55)(59,73)(60,71)(61,72)(62,70)(63,68)
(64,69)(65,76)(66,74)(67,75)(77,80)(78,81)(79,82);;
s4 := ( 6, 7)( 9,10)(12,13)(14,23)(15,25)(16,24)(17,26)(18,28)(19,27)(20,29)
(21,31)(22,30)(33,34)(36,37)(39,40)(41,50)(42,52)(43,51)(44,53)(45,55)(46,54)
(47,56)(48,58)(49,57)(60,61)(63,64)(66,67)(68,77)(69,79)(70,78)(71,80)(72,82)
(73,81)(74,83)(75,85)(76,84);;
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, s1*s2*s1*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*s2*s3*s4*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s2*s4*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(85)!(1,2);
s1 := Sym(85)!(3,4);
s2 := Sym(85)!( 8,11)( 9,12)(10,13)(14,23)(15,24)(16,25)(17,29)(18,30)(19,31)
(20,26)(21,27)(22,28)(32,62)(33,63)(34,64)(35,59)(36,60)(37,61)(38,65)(39,66)
(40,67)(41,80)(42,81)(43,82)(44,77)(45,78)(46,79)(47,83)(48,84)(49,85)(50,71)
(51,72)(52,73)(53,68)(54,69)(55,70)(56,74)(57,75)(58,76);
s3 := Sym(85)!( 5,43)( 6,41)( 7,42)( 8,49)( 9,47)(10,48)(11,46)(12,44)(13,45)
(14,33)(15,34)(16,32)(17,39)(18,40)(19,38)(20,36)(21,37)(22,35)(23,50)(24,51)
(25,52)(26,56)(27,57)(28,58)(29,53)(30,54)(31,55)(59,73)(60,71)(61,72)(62,70)
(63,68)(64,69)(65,76)(66,74)(67,75)(77,80)(78,81)(79,82);
s4 := Sym(85)!( 6, 7)( 9,10)(12,13)(14,23)(15,25)(16,24)(17,26)(18,28)(19,27)
(20,29)(21,31)(22,30)(33,34)(36,37)(39,40)(41,50)(42,52)(43,51)(44,53)(45,55)
(46,54)(47,56)(48,58)(49,57)(60,61)(63,64)(66,67)(68,77)(69,79)(70,78)(71,80)
(72,82)(73,81)(74,83)(75,85)(76,84);
poly := sub<Sym(85)|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, 
s1*s2*s1*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*s2*s3*s4*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s2*s4*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope