Polytope of Type {6,2,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,18}*432
if this polytope has a name.
Group : SmallGroup(432,544)
Rank : 4
Schlafli Type : {6,2,18}
Number of vertices, edges, etc : 6, 6, 18, 18
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,2,18,2} of size 864
   {6,2,18,4} of size 1728
   {6,2,18,4} of size 1728
   {6,2,18,4} of size 1728
Vertex Figure Of :
   {2,6,2,18} of size 864
   {3,6,2,18} of size 1296
   {4,6,2,18} of size 1728
   {3,6,2,18} of size 1728
   {4,6,2,18} of size 1728
   {4,6,2,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,18}*216, {6,2,9}*216
   3-fold quotients : {2,2,18}*144, {6,2,6}*144
   4-fold quotients : {3,2,9}*108
   6-fold quotients : {2,2,9}*72, {3,2,6}*72, {6,2,3}*72
   9-fold quotients : {2,2,6}*48, {6,2,2}*48
   12-fold quotients : {3,2,3}*36
   18-fold quotients : {2,2,3}*24, {3,2,2}*24
   27-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,2,36}*864, {12,2,18}*864, {6,4,18}*864
   3-fold covers : {18,2,18}*1296, {6,6,18}*1296a, {6,2,54}*1296, {6,6,18}*1296b, {6,6,18}*1296c, {6,6,18}*1296e
   4-fold covers : {12,2,36}*1728, {12,4,18}*1728, {6,4,36}*1728, {6,2,72}*1728, {24,2,18}*1728, {6,8,18}*1728, {6,4,18}*1728a, {6,4,18}*1728b
Permutation Representation (GAP) :

s0 := (3,4)(5,6);;
s1 := (1,5)(2,3)(4,6);;
s2 := ( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24);;
s3 := ( 7,11)( 8, 9)(10,15)(12,13)(14,19)(16,17)(18,23)(20,21)(22,24);;
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*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(24)!(3,4)(5,6);
s1 := Sym(24)!(1,5)(2,3)(4,6);
s2 := Sym(24)!( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24);
s3 := Sym(24)!( 7,11)( 8, 9)(10,15)(12,13)(14,19)(16,17)(18,23)(20,21)(22,24);
poly := sub<Sym(24)|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*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

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