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# Polytope of Type {4,15,2}

Atlas Canonical Name : {4,15,2}*240
if this polytope has a name.
Group : SmallGroup(240,197)
Rank : 4
Schlafli Type : {4,15,2}
Number of vertices, edges, etc : 4, 30, 15, 2
Order of s0s1s2s3 : 30
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,15,2,2} of size 480
{4,15,2,3} of size 720
{4,15,2,4} of size 960
{4,15,2,5} of size 1200
{4,15,2,6} of size 1440
{4,15,2,7} of size 1680
{4,15,2,8} of size 1920
Vertex Figure Of :
{2,4,15,2} of size 480
{4,4,15,2} of size 1920
Quotients (Maximal Quotients in Boldface) :
5-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,15,2}*480, {4,30,2}*480b, {4,30,2}*480c
3-fold covers : {4,45,2}*720, {4,15,6}*720
4-fold covers : {4,60,2}*960b, {4,60,2}*960c, {4,30,4}*960c, {8,15,2}*960, {4,30,2}*960, {4,15,4}*960a
5-fold covers : {4,75,2}*1200, {4,15,10}*1200
6-fold covers : {4,45,2}*1440, {4,90,2}*1440b, {4,90,2}*1440c, {4,15,6}*1440b, {4,30,6}*1440d, {4,30,6}*1440e, {4,30,6}*1440f, {12,15,2}*1440, {12,30,2}*1440d
7-fold covers : {4,105,2}*1680
8-fold covers : {4,60,4}*1920d, {4,60,4}*1920e, {4,30,2}*1920a, {8,15,2}*1920a, {8,30,2}*1920a, {4,120,2}*1920c, {4,120,2}*1920d, {4,30,8}*1920b, {4,60,2}*1920b, {4,30,4}*1920b, {4,30,2}*1920b, {4,60,2}*1920c, {8,30,2}*1920b, {8,30,2}*1920c, {4,15,8}*1920, {4,15,4}*1920c, {4,30,4}*1920c, {4,30,4}*1920f
Permutation Representation (GAP) :
```s0 := ( 1, 4)( 2, 6)( 3, 8)( 5,11)( 7,15)( 9,10)(12,16)(13,14)(17,20)(18,19);;
s1 := ( 2, 3)( 4, 9)( 5, 7)( 6,12)( 8,13)(11,17)(14,16)(15,18)(19,20);;
s2 := ( 1, 2)( 3, 5)( 4, 6)( 8,11)( 9,14)(10,13)(12,19)(16,18)(17,20);;
s3 := (21,22);;
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,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s0*s1,
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*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(22)!( 1, 4)( 2, 6)( 3, 8)( 5,11)( 7,15)( 9,10)(12,16)(13,14)(17,20)
(18,19);
s1 := Sym(22)!( 2, 3)( 4, 9)( 5, 7)( 6,12)( 8,13)(11,17)(14,16)(15,18)(19,20);
s2 := Sym(22)!( 1, 2)( 3, 5)( 4, 6)( 8,11)( 9,14)(10,13)(12,19)(16,18)(17,20);
s3 := Sym(22)!(21,22);
poly := sub<Sym(22)|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, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s0*s1, 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*s1*s2 >;

```

to this polytope