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Polytope of Type {6,4}

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

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

References : None.
to this polytope