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Polytope of Type {2,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,6}*432b
if this polytope has a name.
Group : SmallGroup(432,544)
Rank : 4
Schlafli Type : {2,18,6}
Number of vertices, edges, etc : 2, 18, 54, 6
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,18,6,2} of size 864
   {2,18,6,3} of size 1296
   {2,18,6,4} of size 1728
Vertex Figure Of :
   {2,2,18,6} of size 864
   {3,2,18,6} of size 1296
   {4,2,18,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,9,6}*216
   3-fold quotients : {2,18,2}*144, {2,6,6}*144c
   6-fold quotients : {2,9,2}*72, {2,3,6}*72
   9-fold quotients : {2,6,2}*48
   18-fold quotients : {2,3,2}*24
   27-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,36,6}*864b, {4,18,6}*864b, {2,18,12}*864b
   3-fold covers : {2,18,18}*1296c, {2,18,6}*1296a, {2,54,6}*1296b, {6,18,6}*1296c, {6,18,6}*1296d, {2,18,6}*1296i
   4-fold covers : {4,36,6}*1728b, {2,72,6}*1728b, {8,18,6}*1728b, {2,36,12}*1728b, {2,18,24}*1728b, {4,18,12}*1728b, {2,18,6}*1728, {4,18,6}*1728b, {2,18,12}*1728b
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6, 9)( 7,11)( 8,10)(12,22)(13,21)(14,23)(15,28)(16,27)(17,29)
(18,25)(19,24)(20,26)(31,32)(33,36)(34,38)(35,37)(39,49)(40,48)(41,50)(42,55)
(43,54)(44,56)(45,52)(46,51)(47,53);;
s2 := ( 3,42)( 4,44)( 5,43)( 6,39)( 7,41)( 8,40)( 9,45)(10,47)(11,46)(12,33)
(13,35)(14,34)(15,30)(16,32)(17,31)(18,36)(19,38)(20,37)(21,52)(22,51)(23,53)
(24,49)(25,48)(26,50)(27,55)(28,54)(29,56);;
s3 := ( 6, 9)( 7,10)( 8,11)(15,18)(16,19)(17,20)(24,27)(25,28)(26,29)(33,36)
(34,37)(35,38)(42,45)(43,46)(44,47)(51,54)(52,55)(53,56);;
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*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(56)!(1,2);
s1 := Sym(56)!( 4, 5)( 6, 9)( 7,11)( 8,10)(12,22)(13,21)(14,23)(15,28)(16,27)
(17,29)(18,25)(19,24)(20,26)(31,32)(33,36)(34,38)(35,37)(39,49)(40,48)(41,50)
(42,55)(43,54)(44,56)(45,52)(46,51)(47,53);
s2 := Sym(56)!( 3,42)( 4,44)( 5,43)( 6,39)( 7,41)( 8,40)( 9,45)(10,47)(11,46)
(12,33)(13,35)(14,34)(15,30)(16,32)(17,31)(18,36)(19,38)(20,37)(21,52)(22,51)
(23,53)(24,49)(25,48)(26,50)(27,55)(28,54)(29,56);
s3 := Sym(56)!( 6, 9)( 7,10)( 8,11)(15,18)(16,19)(17,20)(24,27)(25,28)(26,29)
(33,36)(34,37)(35,38)(42,45)(43,46)(44,47)(51,54)(52,55)(53,56);
poly := sub<Sym(56)|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*s1*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 

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