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Polytope of Type {2,18,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,9}*1944c
if this polytope has a name.
Group : SmallGroup(1944,944)
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, {2,18,3}*648
   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)(30,58)(31,57)(32,59)(33,61)(34,60)(35,62)(36,64)(37,63)(38,65)
(39,79)(40,78)(41,80)(42,82)(43,81)(44,83)(45,76)(46,75)(47,77)(48,73)(49,72)
(50,74)(51,67)(52,66)(53,68)(54,70)(55,69)(56,71);;
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)(57,58)(60,64)(61,63)(62,65)(67,68)
(69,72)(70,74)(71,73)(75,77)(78,83)(79,82)(80,81);;
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)(30,79)(31,78)(32,80)(33,76)(34,75)(35,77)(36,82)(37,81)(38,83)
(39,58)(40,57)(41,59)(42,64)(43,63)(44,65)(45,61)(46,60)(47,62)(48,73)(49,72)
(50,74)(51,70)(52,69)(53,71)(54,67)(55,66)(56,68);;
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, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*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, 
s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
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)(30,58)(31,57)(32,59)(33,61)(34,60)(35,62)(36,64)(37,63)
(38,65)(39,79)(40,78)(41,80)(42,82)(43,81)(44,83)(45,76)(46,75)(47,77)(48,73)
(49,72)(50,74)(51,67)(52,66)(53,68)(54,70)(55,69)(56,71);
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)(57,58)(60,64)(61,63)(62,65)
(67,68)(69,72)(70,74)(71,73)(75,77)(78,83)(79,82)(80,81);
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)(30,79)(31,78)(32,80)(33,76)(34,75)(35,77)(36,82)(37,81)
(38,83)(39,58)(40,57)(41,59)(42,64)(43,63)(44,65)(45,61)(46,60)(47,62)(48,73)
(49,72)(50,74)(51,70)(52,69)(53,71)(54,67)(55,66)(56,68);
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, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*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, 
s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 

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