# Polytope of Type {4,4,10}

Atlas Canonical Name : {4,4,10}*640
Also Known As : {{4,4}4,{4,10|2}}. if this polytope has another name.
Group : SmallGroup(640,14119)
Rank : 4
Schlafli Type : {4,4,10}
Number of vertices, edges, etc : 8, 16, 40, 10
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,4,10,2} of size 1280
Vertex Figure Of :
{2,4,4,10} of size 1280
{3,4,4,10} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4,10}*320
4-fold quotients : {2,4,10}*160, {4,2,10}*160
5-fold quotients : {4,4,2}*128
8-fold quotients : {4,2,5}*80, {2,2,10}*80
10-fold quotients : {4,4,2}*64
16-fold quotients : {2,2,5}*40
20-fold quotients : {2,4,2}*32, {4,2,2}*32
40-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,8,10}*1280a, {8,4,10}*1280a, {4,4,20}*1280a, {4,4,10}*1280, {4,8,10}*1280b, {8,4,10}*1280b
3-fold covers : {4,4,30}*1920a, {4,12,10}*1920a, {12,4,10}*1920a
Permutation Representation (GAP) :
```s0 := (11,16)(12,17)(13,18)(14,19)(15,20)(31,36)(32,37)(33,38)(34,39)(35,40);;
s1 := (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40);;
s2 := ( 1,21)( 2,25)( 3,24)( 4,23)( 5,22)( 6,26)( 7,30)( 8,29)( 9,28)(10,27)
(11,31)(12,35)(13,34)(14,33)(15,32)(16,36)(17,40)(18,39)(19,38)(20,37);;
s3 := ( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)(23,25)
(26,27)(28,30)(31,32)(33,35)(36,37)(38,40);;
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, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(40)!(11,16)(12,17)(13,18)(14,19)(15,20)(31,36)(32,37)(33,38)(34,39)
(35,40);
s1 := Sym(40)!(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)
(30,40);
s2 := Sym(40)!( 1,21)( 2,25)( 3,24)( 4,23)( 5,22)( 6,26)( 7,30)( 8,29)( 9,28)
(10,27)(11,31)(12,35)(13,34)(14,33)(15,32)(16,36)(17,40)(18,39)(19,38)(20,37);
s3 := Sym(40)!( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)
(23,25)(26,27)(28,30)(31,32)(33,35)(36,37)(38,40);
poly := sub<Sym(40)|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,
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope