Polytope of Type {3,2,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,18,6}*1944a
if this polytope has a name.
Group : SmallGroup(1944,2340)
Rank : 5
Schlafli Type : {3,2,18,6}
Number of vertices, edges, etc : 3, 3, 27, 81, 9
Order of s₀s₁s₂s₃s₄ : 9
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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 : {3,2,6,6}*648
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

to this polytope