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# Polytope of Type {6,2,2,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,2,6}*288
if this polytope has a name.
Group : SmallGroup(288,1040)
Rank : 5
Schlafli Type : {6,2,2,6}
Number of vertices, edges, etc : 6, 6, 2, 6, 6
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,2,2,6,2} of size 576
{6,2,2,6,3} of size 864
{6,2,2,6,4} of size 1152
{6,2,2,6,3} of size 1152
{6,2,2,6,4} of size 1152
{6,2,2,6,4} of size 1152
{6,2,2,6,4} of size 1728
{6,2,2,6,6} of size 1728
{6,2,2,6,6} of size 1728
{6,2,2,6,6} of size 1728
Vertex Figure Of :
{2,6,2,2,6} of size 576
{3,6,2,2,6} of size 864
{4,6,2,2,6} of size 1152
{3,6,2,2,6} of size 1152
{4,6,2,2,6} of size 1152
{4,6,2,2,6} of size 1152
{4,6,2,2,6} of size 1728
{6,6,2,2,6} of size 1728
{6,6,2,2,6} of size 1728
{6,6,2,2,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,2,2,6}*144, {6,2,2,3}*144
3-fold quotients : {2,2,2,6}*96, {6,2,2,2}*96
4-fold quotients : {3,2,2,3}*72
6-fold quotients : {2,2,2,3}*48, {3,2,2,2}*48
9-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,2,2,12}*576, {12,2,2,6}*576, {6,2,4,6}*576a, {6,4,2,6}*576a
3-fold covers : {6,2,2,18}*864, {18,2,2,6}*864, {6,2,6,6}*864a, {6,2,6,6}*864b, {6,6,2,6}*864a, {6,6,2,6}*864c
4-fold covers : {6,4,4,6}*1152, {6,2,4,12}*1152a, {12,4,2,6}*1152a, {6,4,2,12}*1152a, {12,2,4,6}*1152a, {12,2,2,12}*1152, {6,2,8,6}*1152, {6,8,2,6}*1152, {6,2,2,24}*1152, {24,2,2,6}*1152, {6,2,4,6}*1152, {6,4,2,6}*1152
5-fold covers : {6,2,10,6}*1440, {6,10,2,6}*1440, {6,2,2,30}*1440, {30,2,2,6}*1440
6-fold covers : {12,2,2,18}*1728, {18,2,2,12}*1728, {6,2,2,36}*1728, {36,2,2,6}*1728, {6,2,4,18}*1728a, {6,4,2,18}*1728a, {18,2,4,6}*1728a, {18,4,2,6}*1728a, {6,2,6,12}*1728a, {6,2,6,12}*1728b, {6,2,12,6}*1728a, {6,6,2,12}*1728a, {6,6,2,12}*1728c, {6,12,2,6}*1728a, {12,2,6,6}*1728a, {12,2,6,6}*1728b, {12,6,2,6}*1728a, {12,6,2,6}*1728b, {6,4,6,6}*1728a, {6,6,4,6}*1728a, {6,4,6,6}*1728c, {6,6,4,6}*1728c, {6,2,12,6}*1728c, {6,12,2,6}*1728c
Permutation Representation (GAP) :
s0 := (3,4)(5,6);;
s1 := (1,5)(2,3)(4,6);;
s2 := (7,8);;
s3 := (11,12)(13,14);;
s4 := ( 9,13)(10,11)(12,14);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(14)!(3,4)(5,6);
s1 := Sym(14)!(1,5)(2,3)(4,6);
s2 := Sym(14)!(7,8);
s3 := Sym(14)!(11,12)(13,14);
s4 := Sym(14)!( 9,13)(10,11)(12,14);
poly := sub<Sym(14)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

to this polytope