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}*1944a
if this polytope has a name.
Group : SmallGroup(1944,2340)
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 : {3,2,9,6}*648, {3,2,3,6}*648
   9-fold quotients : {3,2,9,2}*216, {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 := ( 7,10)( 8,11)( 9,12)(13,22)(14,23)(15,24)(16,28)(17,29)(18,30)(19,25)
(20,26)(21,27)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,64)(38,65)(39,66)
(40,79)(41,80)(42,81)(43,76)(44,77)(45,78)(46,82)(47,83)(48,84)(49,70)(50,71)
(51,72)(52,67)(53,68)(54,69)(55,73)(56,74)(57,75);;
s3 := ( 4,42)( 5,40)( 6,41)( 7,48)( 8,46)( 9,47)(10,45)(11,43)(12,44)(13,32)
(14,33)(15,31)(16,38)(17,39)(18,37)(19,35)(20,36)(21,34)(22,49)(23,50)(24,51)
(25,55)(26,56)(27,57)(28,52)(29,53)(30,54)(58,72)(59,70)(60,71)(61,69)(62,67)
(63,68)(64,75)(65,73)(66,74)(76,79)(77,80)(78,81);;
s4 := ( 5, 6)( 8, 9)(11,12)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26)(19,28)
(20,30)(21,29)(32,33)(35,36)(38,39)(40,49)(41,51)(42,50)(43,52)(44,54)(45,53)
(46,55)(47,57)(48,56)(59,60)(62,63)(65,66)(67,76)(68,78)(69,77)(70,79)(71,81)
(72,80)(73,82)(74,84)(75,83);;
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, 
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(84)!(2,3);
s1 := Sym(84)!(1,2);
s2 := Sym(84)!( 7,10)( 8,11)( 9,12)(13,22)(14,23)(15,24)(16,28)(17,29)(18,30)
(19,25)(20,26)(21,27)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,64)(38,65)
(39,66)(40,79)(41,80)(42,81)(43,76)(44,77)(45,78)(46,82)(47,83)(48,84)(49,70)
(50,71)(51,72)(52,67)(53,68)(54,69)(55,73)(56,74)(57,75);
s3 := Sym(84)!( 4,42)( 5,40)( 6,41)( 7,48)( 8,46)( 9,47)(10,45)(11,43)(12,44)
(13,32)(14,33)(15,31)(16,38)(17,39)(18,37)(19,35)(20,36)(21,34)(22,49)(23,50)
(24,51)(25,55)(26,56)(27,57)(28,52)(29,53)(30,54)(58,72)(59,70)(60,71)(61,69)
(62,67)(63,68)(64,75)(65,73)(66,74)(76,79)(77,80)(78,81);
s4 := Sym(84)!( 5, 6)( 8, 9)(11,12)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26)
(19,28)(20,30)(21,29)(32,33)(35,36)(38,39)(40,49)(41,51)(42,50)(43,52)(44,54)
(45,53)(46,55)(47,57)(48,56)(59,60)(62,63)(65,66)(67,76)(68,78)(69,77)(70,79)
(71,81)(72,80)(73,82)(74,84)(75,83);
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, 
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