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# Polytope of Type {2,8,4}

Atlas Canonical Name : {2,8,4}*256b
if this polytope has a name.
Group : SmallGroup(256,26531)
Rank : 4
Schlafli Type : {2,8,4}
Number of vertices, edges, etc : 2, 16, 32, 8
Order of s0s1s2s3 : 4
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,8,4,2} of size 512
Vertex Figure Of :
{2,2,8,4} of size 512
{3,2,8,4} of size 768
{5,2,8,4} of size 1280
{7,2,8,4} of size 1792
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,4,4}*128
4-fold quotients : {2,4,4}*64
8-fold quotients : {2,2,4}*32, {2,4,2}*32
16-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,8,4}*512a, {2,8,8}*512b, {2,8,8}*512c, {4,8,4}*512h, {2,8,4}*512b, {2,8,4}*512d, {2,8,8}*512f, {2,8,8}*512h
3-fold covers : {6,8,4}*768b, {2,24,4}*768b, {2,8,12}*768b
5-fold covers : {10,8,4}*1280b, {2,40,4}*1280b, {2,8,20}*1280b
7-fold covers : {14,8,4}*1792b, {2,56,4}*1792b, {2,8,28}*1792b
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,11)( 4,12)( 5,13)( 6,14)( 7,17)( 8,18)( 9,15)(10,16);;
s2 := ( 5, 6)( 9,10)(11,15)(12,16)(13,18)(14,17);;
s3 := ( 7, 9)( 8,10)(15,17)(16,18);;
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*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*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(18)!(1,2);
s1 := Sym(18)!( 3,11)( 4,12)( 5,13)( 6,14)( 7,17)( 8,18)( 9,15)(10,16);
s2 := Sym(18)!( 5, 6)( 9,10)(11,15)(12,16)(13,18)(14,17);
s3 := Sym(18)!( 7, 9)( 8,10)(15,17)(16,18);
poly := sub<Sym(18)|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*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2 >;

to this polytope