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# Polytope of Type {6,4,10}

Atlas Canonical Name : {6,4,10}*720
if this polytope has a name.
Group : SmallGroup(720,784)
Rank : 4
Schlafli Type : {6,4,10}
Number of vertices, edges, etc : 9, 18, 30, 10
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,4,10,2} of size 1440
Vertex Figure Of :
{2,6,4,10} of size 1440
Quotients (Maximal Quotients in Boldface) :
5-fold quotients : {6,4,2}*144
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,4,20}*1440, {6,4,10}*1440c
Permutation Representation (GAP) :
```s0 := ( 6,11)( 7,12)( 8,13)( 9,14)(10,15)(16,31)(17,32)(18,33)(19,34)(20,35)
(21,41)(22,42)(23,43)(24,44)(25,45)(26,36)(27,37)(28,38)(29,39)(30,40);;
s1 := ( 1,16)( 2,17)( 3,18)( 4,19)( 5,20)(11,41)(12,42)(13,43)(14,44)(15,45)
(21,36)(22,37)(23,38)(24,39)(25,40);;
s2 := ( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(16,41)(17,45)(18,44)(19,43)
(20,42)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)
(30,37);;
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)(41,42)(43,45);;
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, 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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*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(45)!( 6,11)( 7,12)( 8,13)( 9,14)(10,15)(16,31)(17,32)(18,33)(19,34)
(20,35)(21,41)(22,42)(23,43)(24,44)(25,45)(26,36)(27,37)(28,38)(29,39)(30,40);
s1 := Sym(45)!( 1,16)( 2,17)( 3,18)( 4,19)( 5,20)(11,41)(12,42)(13,43)(14,44)
(15,45)(21,36)(22,37)(23,38)(24,39)(25,40);
s2 := Sym(45)!( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(16,41)(17,45)(18,44)
(19,43)(20,42)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)
(30,37);
s3 := Sym(45)!( 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)(41,42)(43,45);
poly := sub<Sym(45)|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,
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*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