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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6,2,2}*864b
if this polytope has a name.
Group : SmallGroup(864,4032)
Rank : 5
Schlafli Type : {18,6,2,2}
Number of vertices, edges, etc : 18, 54, 6, 2, 2
Order of s0s1s2s3s4 : 18
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {18,6,2,2,2} of size 1728
Vertex Figure Of :
   {2,18,6,2,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {9,6,2,2}*432
   3-fold quotients : {18,2,2,2}*288, {6,6,2,2}*288c
   6-fold quotients : {9,2,2,2}*144, {3,6,2,2}*144
   9-fold quotients : {6,2,2,2}*96
   18-fold quotients : {3,2,2,2}*48
   27-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,6,2,2}*1728b, {18,6,2,4}*1728b, {18,6,4,2}*1728b, {18,12,2,2}*1728b
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,20)(11,19)(12,21)(13,26)(14,25)(15,27)
(16,23)(17,22)(18,24)(29,30)(31,34)(32,36)(33,35)(37,47)(38,46)(39,48)(40,53)
(41,52)(42,54)(43,50)(44,49)(45,51);;
s1 := ( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,43)( 8,45)( 9,44)(10,31)
(11,33)(12,32)(13,28)(14,30)(15,29)(16,34)(17,36)(18,35)(19,50)(20,49)(21,51)
(22,47)(23,46)(24,48)(25,53)(26,52)(27,54);;
s2 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54);;
s3 := (55,56);;
s4 := (57,58);;
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, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(58)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,20)(11,19)(12,21)(13,26)(14,25)
(15,27)(16,23)(17,22)(18,24)(29,30)(31,34)(32,36)(33,35)(37,47)(38,46)(39,48)
(40,53)(41,52)(42,54)(43,50)(44,49)(45,51);
s1 := Sym(58)!( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,43)( 8,45)( 9,44)
(10,31)(11,33)(12,32)(13,28)(14,30)(15,29)(16,34)(17,36)(18,35)(19,50)(20,49)
(21,51)(22,47)(23,46)(24,48)(25,53)(26,52)(27,54);
s2 := Sym(58)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54);
s3 := Sym(58)!(55,56);
s4 := Sym(58)!(57,58);
poly := sub<Sym(58)|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, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 

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