Polytope of Type {6,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,3}*648c
Also Known As : {{6,6|2},{6,3}6}. if this polytope has another name.
Group : SmallGroup(648,555)
Rank : 4
Schlafli Type : {6,6,3}
Number of vertices, edges, etc : 6, 54, 27, 9
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,6,3,2} of size 1296
Vertex Figure Of :
{2,6,6,3} of size 1296
{3,6,6,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,6,3}*216, {6,6,3}*216b
9-fold quotients : {2,6,3}*72, {6,2,3}*72
18-fold quotients : {3,2,3}*36
27-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,6,3}*1296e, {6,6,6}*1296f
3-fold covers : {6,6,9}*1944c, {18,6,3}*1944e, {6,6,3}*1944b, {6,6,3}*1944c, {6,6,9}*1944f, {6,6,9}*1944g, {6,6,9}*1944h, {6,6,3}*1944h, {6,18,3}*1944
Permutation Representation (GAP) :
s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)(20,24)
(21,23)(26,27);;
s2 := (10,20)(11,21)(12,19)(13,23)(14,24)(15,22)(16,26)(17,27)(18,25);;
s3 := ( 1,10)( 2,11)( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18);;
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, s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(27)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27);
s1 := Sym(27)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)
(20,24)(21,23)(26,27);
s2 := Sym(27)!(10,20)(11,21)(12,19)(13,23)(14,24)(15,22)(16,26)(17,27)(18,25);
s3 := Sym(27)!( 1,10)( 2,11)( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18);
poly := sub<Sym(27)|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,
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2 >;
References : None.
to this polytope