Polytope of Type {3,2,9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,9,6}*1944d
if this polytope has a name.
Group : SmallGroup(1944,2345)
Rank : 5
Schlafli Type : {3,2,9,6}
Number of vertices, edges, etc : 3, 3, 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 : {3,2,3,6}*648
   9-fold quotients : {3,2,3,6}*216
   27-fold quotients : {3,2,3,2}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

to this polytope