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# Polytope of Type {2,12,4}

Atlas Canonical Name : {2,12,4}*576
if this polytope has a name.
Group : SmallGroup(576,8418)
Rank : 4
Schlafli Type : {2,12,4}
Number of vertices, edges, etc : 2, 36, 72, 12
Order of s0s1s2s3 : 4
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,12,4,2} of size 1152
Vertex Figure Of :
{2,2,12,4} of size 1152
{3,2,12,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,6,4}*288
4-fold quotients : {2,6,4}*144
9-fold quotients : {2,4,4}*64
18-fold quotients : {2,2,4}*32, {2,4,2}*32
36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12,4}*1152a, {2,24,4}*1152a, {2,12,8}*1152a, {2,24,4}*1152b, {2,12,8}*1152b, {2,12,4}*1152
3-fold covers : {2,12,4}*1728b, {2,12,12}*1728f, {2,12,12}*1728g, {2,12,4}*1728d, {2,12,12}*1728j, {6,12,4}*1728n, {2,12,12}*1728l
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 7, 8)( 9,12)(10,14)(11,13);;
s2 := ( 3,10)( 4, 9)( 5,11)( 6,13)( 7,12)( 8,14);;
s3 := (10,11)(13,14);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

to this polytope