Polytope of Type {2,18,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,9}*1944b
if this polytope has a name.
Group : SmallGroup(1944,943)
Rank : 4
Schlafli Type : {2,18,9}
Number of vertices, edges, etc : 2, 54, 243, 27
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 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 : {2,6,9}*648c
   9-fold quotients : {2,6,3}*216
   27-fold quotients : {2,6,3}*72
   81-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s3*s1*s2*s3*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s3*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s1*s2*s1*s3*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s3*s1*s2*s3*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s3*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s1*s2*s1*s3*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s3 >;

to this polytope