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# Polytope of Type {10,2,2,2}

Atlas Canonical Name : {10,2,2,2}*160
if this polytope has a name.
Group : SmallGroup(160,237)
Rank : 5
Schlafli Type : {10,2,2,2}
Number of vertices, edges, etc : 10, 10, 2, 2, 2
Order of s0s1s2s3s4 : 10
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{10,2,2,2,2} of size 320
{10,2,2,2,3} of size 480
{10,2,2,2,4} of size 640
{10,2,2,2,5} of size 800
{10,2,2,2,6} of size 960
{10,2,2,2,7} of size 1120
{10,2,2,2,8} of size 1280
{10,2,2,2,9} of size 1440
{10,2,2,2,10} of size 1600
{10,2,2,2,11} of size 1760
{10,2,2,2,12} of size 1920
Vertex Figure Of :
{2,10,2,2,2} of size 320
{4,10,2,2,2} of size 640
{5,10,2,2,2} of size 800
{3,10,2,2,2} of size 960
{3,10,2,2,2} of size 960
{5,10,2,2,2} of size 960
{5,10,2,2,2} of size 960
{6,10,2,2,2} of size 960
{8,10,2,2,2} of size 1280
{4,10,2,2,2} of size 1600
{10,10,2,2,2} of size 1600
{10,10,2,2,2} of size 1600
{10,10,2,2,2} of size 1600
{12,10,2,2,2} of size 1920
{4,10,2,2,2} of size 1920
{4,10,2,2,2} of size 1920
{6,10,2,2,2} of size 1920
{6,10,2,2,2} of size 1920
{3,10,2,2,2} of size 1920
{5,10,2,2,2} of size 1920
{6,10,2,2,2} of size 1920
{6,10,2,2,2} of size 1920
{6,10,2,2,2} of size 1920
{6,10,2,2,2} of size 1920
{10,10,2,2,2} of size 1920
{10,10,2,2,2} of size 1920
{10,10,2,2,2} of size 1920
{10,10,2,2,2} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {5,2,2,2}*80
5-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
2-fold covers : {20,2,2,2}*320, {10,2,2,4}*320, {10,2,4,2}*320, {10,4,2,2}*320
3-fold covers : {10,2,2,6}*480, {10,2,6,2}*480, {10,6,2,2}*480, {30,2,2,2}*480
4-fold covers : {20,4,2,2}*640, {20,2,2,4}*640, {20,2,4,2}*640, {10,2,4,4}*640, {10,4,4,2}*640, {10,4,2,4}*640, {40,2,2,2}*640, {10,2,2,8}*640, {10,2,8,2}*640, {10,8,2,2}*640
5-fold covers : {50,2,2,2}*800, {10,2,2,10}*800, {10,2,10,2}*800, {10,10,2,2}*800a, {10,10,2,2}*800c
6-fold covers : {10,2,2,12}*960, {10,2,12,2}*960, {10,12,2,2}*960, {20,2,2,6}*960, {20,2,6,2}*960, {20,6,2,2}*960a, {10,2,4,6}*960a, {10,2,6,4}*960a, {10,4,2,6}*960, {10,4,6,2}*960, {10,6,2,4}*960, {10,6,4,2}*960a, {60,2,2,2}*960, {30,2,2,4}*960, {30,2,4,2}*960, {30,4,2,2}*960a
7-fold covers : {10,2,2,14}*1120, {10,2,14,2}*1120, {10,14,2,2}*1120, {70,2,2,2}*1120
8-fold covers : {10,4,4,4}*1280, {20,4,4,2}*1280, {20,2,4,4}*1280, {20,4,2,4}*1280, {10,2,4,8}*1280a, {10,2,8,4}*1280a, {10,4,8,2}*1280a, {10,8,4,2}*1280a, {20,8,2,2}*1280a, {40,4,2,2}*1280a, {10,2,4,8}*1280b, {10,2,8,4}*1280b, {10,4,8,2}*1280b, {10,8,4,2}*1280b, {20,8,2,2}*1280b, {40,4,2,2}*1280b, {10,2,4,4}*1280, {10,4,4,2}*1280, {20,4,2,2}*1280, {10,4,2,8}*1280, {10,8,2,4}*1280, {20,2,2,8}*1280, {20,2,8,2}*1280, {40,2,2,4}*1280, {40,2,4,2}*1280, {10,2,2,16}*1280, {10,2,16,2}*1280, {10,16,2,2}*1280, {80,2,2,2}*1280
9-fold covers : {10,2,2,18}*1440, {10,2,18,2}*1440, {10,18,2,2}*1440, {90,2,2,2}*1440, {10,2,6,6}*1440a, {10,2,6,6}*1440b, {10,2,6,6}*1440c, {10,6,2,6}*1440, {10,6,6,2}*1440a, {10,6,6,2}*1440b, {10,6,6,2}*1440c, {30,6,2,2}*1440a, {30,2,2,6}*1440, {30,2,6,2}*1440, {30,6,2,2}*1440b, {30,6,2,2}*1440c
10-fold covers : {100,2,2,2}*1600, {50,2,2,4}*1600, {50,2,4,2}*1600, {50,4,2,2}*1600, {10,2,2,20}*1600, {10,2,20,2}*1600, {10,20,2,2}*1600a, {20,2,2,10}*1600, {20,2,10,2}*1600, {20,10,2,2}*1600a, {20,10,2,2}*1600b, {10,2,4,10}*1600, {10,2,10,4}*1600, {10,4,2,10}*1600, {10,4,10,2}*1600, {10,10,2,4}*1600a, {10,10,2,4}*1600c, {10,10,4,2}*1600a, {10,10,4,2}*1600c, {10,20,2,2}*1600c
11-fold covers : {10,2,2,22}*1760, {10,2,22,2}*1760, {10,22,2,2}*1760, {110,2,2,2}*1760
12-fold covers : {30,2,4,4}*1920, {30,4,4,2}*1920, {60,4,2,2}*1920a, {10,4,4,6}*1920, {10,6,4,4}*1920, {10,2,4,12}*1920a, {10,2,12,4}*1920a, {10,4,12,2}*1920, {10,12,4,2}*1920a, {20,4,2,6}*1920, {20,4,6,2}*1920, {20,12,2,2}*1920, {30,4,2,4}*1920a, {60,2,2,4}*1920, {60,2,4,2}*1920, {10,4,6,4}*1920a, {10,4,2,12}*1920, {10,12,2,4}*1920, {20,2,4,6}*1920a, {20,2,6,4}*1920a, {20,6,2,4}*1920a, {20,6,4,2}*1920a, {20,2,2,12}*1920, {20,2,12,2}*1920, {30,2,2,8}*1920, {30,2,8,2}*1920, {30,8,2,2}*1920, {120,2,2,2}*1920, {10,2,6,8}*1920, {10,2,8,6}*1920, {10,6,2,8}*1920, {10,6,8,2}*1920, {10,8,2,6}*1920, {10,8,6,2}*1920, {10,2,2,24}*1920, {10,2,24,2}*1920, {10,24,2,2}*1920, {40,2,2,6}*1920, {40,2,6,2}*1920, {40,6,2,2}*1920, {10,2,4,6}*1920, {10,2,6,4}*1920, {10,2,6,6}*1920, {10,4,6,2}*1920, {10,6,4,2}*1920a, {10,6,6,2}*1920, {20,6,2,2}*1920a, {30,6,2,2}*1920, {30,4,2,2}*1920
Permutation Representation (GAP) :
```s0 := ( 3, 4)( 5, 6)( 7, 8)( 9,10);;
s1 := ( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,10);;
s2 := (11,12);;
s3 := (13,14);;
s4 := (15,16);;
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, s3*s4*s3*s4, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(16)!( 3, 4)( 5, 6)( 7, 8)( 9,10);
s1 := Sym(16)!( 1, 5)( 2, 3)( 4, 9)( 6, 7)( 8,10);
s2 := Sym(16)!(11,12);
s3 := Sym(16)!(13,14);
s4 := Sym(16)!(15,16);
poly := sub<Sym(16)|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,
s3*s4*s3*s4, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```

to this polytope