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# Polytope of Type {2,3,5}

Atlas Canonical Name : {2,3,5}*120
if this polytope has a name.
Group : SmallGroup(120,35)
Rank : 4
Schlafli Type : {2,3,5}
Number of vertices, edges, etc : 2, 6, 15, 10
Order of s0s1s2s3 : 10
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Locally Projective
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,3,5,2} of size 240
{2,3,5,3} of size 1320
{2,3,5,4} of size 1920
Vertex Figure Of :
{2,2,3,5} of size 240
{3,2,3,5} of size 360
{4,2,3,5} of size 480
{5,2,3,5} of size 600
{6,2,3,5} of size 720
{7,2,3,5} of size 840
{8,2,3,5} of size 960
{9,2,3,5} of size 1080
{10,2,3,5} of size 1200
{11,2,3,5} of size 1320
{12,2,3,5} of size 1440
{13,2,3,5} of size 1560
{14,2,3,5} of size 1680
{15,2,3,5} of size 1800
{16,2,3,5} of size 1920
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,3,5}*240, {2,3,10}*240a, {2,3,10}*240b, {2,6,5}*240b, {2,6,5}*240c
4-fold covers : {4,6,5}*480b, {2,3,10}*480, {2,6,5}*480b, {2,6,10}*480c, {2,6,10}*480d, {2,6,10}*480e, {2,6,10}*480f
6-fold covers : {2,3,10}*720, {6,6,5}*720b, {2,3,15}*720, {2,6,15}*720
8-fold covers : {8,6,5}*960b, {4,6,5}*960b, {4,6,10}*960c, {4,6,10}*960d, {2,6,20}*960a, {2,6,20}*960b, {2,12,10}*960c, {2,12,10}*960d, {2,3,20}*960, {2,12,5}*960, {2,6,10}*960c
10-fold covers : {2,6,5}*1200, {2,15,5}*1200, {10,6,5}*1200b, {2,15,10}*1200
12-fold covers : {12,6,5}*1440b, {6,6,5}*1440b, {6,6,10}*1440c, {6,6,10}*1440d, {2,3,10}*1440b, {2,3,30}*1440, {2,6,10}*1440b, {2,6,10}*1440c, {2,6,15}*1440c, {2,6,15}*1440d, {2,6,30}*1440a, {2,6,30}*1440b
14-fold covers : {2,6,35}*1680, {2,21,10}*1680, {14,6,5}*1680b
16-fold covers : {16,6,5}*1920b, {4,12,10}*1920f, {4,12,10}*1920g, {8,6,5}*1920b, {2,6,40}*1920d, {8,6,10}*1920e, {2,6,40}*1920e, {8,6,10}*1920f, {2,24,10}*1920c, {2,24,10}*1920d, {4,6,10}*1920d, {2,6,20}*1920c, {2,12,10}*1920c, {4,12,5}*1920, {2,6,20}*1920e, {2,12,10}*1920e, {2,6,10}*1920b, {4,3,5}*1920, {2,6,5}*1920
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (4,5)(6,7);;
s2 := (3,4)(6,7);;
s3 := (4,6)(5,7);;
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,
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s1*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(7)!(1,2);
s1 := Sym(7)!(4,5)(6,7);
s2 := Sym(7)!(3,4)(6,7);
s3 := Sym(7)!(4,6)(5,7);
poly := sub<Sym(7)|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, s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s1*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3 >;

```

to this polytope