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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,18}*1944c
if this polytope has a name.
Group : SmallGroup(1944,2346)
Rank : 5
Schlafli Type : {3,2,6,18}
Number of vertices, edges, etc : 3, 3, 9, 81, 27
Order of s₀s₁s₂s₃s₄ : 3
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 := ( 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);;
s3 := ( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(10,19)(11,21)(12,20)(23,24)
(26,27)(29,30)(31,69)(32,68)(33,67)(34,72)(35,71)(36,70)(37,75)(38,74)(39,73)
(40,60)(41,59)(42,58)(43,63)(44,62)(45,61)(46,66)(47,65)(48,64)(49,78)(50,77)
(51,76)(52,81)(53,80)(54,79)(55,84)(56,83)(57,82);;
s4 := ( 4,31)( 5,33)( 6,32)( 7,37)( 8,39)( 9,38)(10,34)(11,36)(12,35)(13,43)
(14,45)(15,44)(16,40)(17,42)(18,41)(19,46)(20,48)(21,47)(22,56)(23,55)(24,57)
(25,53)(26,52)(27,54)(28,50)(29,49)(30,51)(58,60)(61,66)(62,65)(63,64)(67,72)
(68,71)(69,70)(73,75)(76,82)(77,84)(78,83)(80,81);;
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*s2*s3*s4*s2*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope