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# Polytope of Type {10,4,4}

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

```
References : None.
to this polytope