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# Polytope of Type {4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6}*216
if this polytope has a name.
Group : SmallGroup(216,87)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 18, 54, 27
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
{4,6,2} of size 432
Vertex Figure Of :
{2,4,6} of size 432
{4,4,6} of size 864
{6,4,6} of size 1296
{8,4,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,6}*432a
3-fold covers : {12,6}*648
4-fold covers : {8,6}*864a, {4,12}*864b
5-fold covers : {20,6}*1080
6-fold covers : {4,6}*1296a, {12,6}*1296l, {12,6}*1296o
7-fold covers : {28,6}*1512
8-fold covers : {16,6}*1728a, {4,12}*1728a, {8,12}*1728a, {4,24}*1728a, {4,24}*1728c, {8,12}*1728d
9-fold covers : {4,18}*1944a, {4,6}*1944, {4,18}*1944b, {4,18}*1944c, {36,6}*1944
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(13,16)(14,18)(15,17);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17);;
s2 := ( 1, 4)( 2, 5)( 3, 6)(13,18)(14,16)(15,17);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0 ];;
poly := F / rels;;

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

Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0 >;

References : None.
to this polytope