# Polytope of Type {6,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,6}*432b
if this polytope has a name.
Group : SmallGroup(432,741)
Rank : 4
Schlafli Type : {6,4,6}
Number of vertices, edges, etc : 6, 18, 18, 9
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,4,6,2} of size 864
Vertex Figure Of :
{2,6,4,6} of size 864
{3,6,4,6} of size 1296
{4,6,4,6} of size 1728
{3,6,4,6} of size 1728
{4,6,4,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,4,6}*144
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,4,6}*864, {6,4,6}*864a
3-fold covers : {18,4,6}*1296, {6,4,6}*1296b, {6,12,6}*1296c, {6,12,6}*1296d, {6,12,6}*1296f, {6,12,6}*1296i
4-fold covers : {24,4,6}*1728, {6,8,6}*1728b, {12,4,6}*1728a, {6,4,12}*1728b
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18);;
s1 := ( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,11)(13,14)(16,17);;
s2 := ( 1,10)( 2,11)( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18);;
s3 := ( 4, 7)( 5, 8)( 6, 9)(10,16)(11,17)(12,18);;
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,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
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*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1 ];;
poly := F / rels;;

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

References : None.
to this polytope