# Polytope of Type {10,4,6}

Atlas Canonical Name : {10,4,6}*1440c
Also Known As : {{10,4|2},{4,6}4}. if this polytope has another name.
Group : SmallGroup(1440,5890)
Rank : 4
Schlafli Type : {10,4,6}
Number of vertices, edges, etc : 10, 60, 36, 18
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {10,4,6}*720
5-fold quotients : {2,4,6}*288
9-fold quotients : {10,4,2}*160
10-fold quotients : {2,4,6}*144
18-fold quotients : {10,2,2}*80
36-fold quotients : {5,2,2}*40
45-fold quotients : {2,4,2}*32
90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
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)(42,45)(43,44)(47,50)(48,49)(52,55)
(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79)
(82,85)(83,84)(87,90)(88,89);;
s1 := ( 1, 2)( 3, 5)( 6,37)( 7,36)( 8,40)( 9,39)(10,38)(11,27)(12,26)(13,30)
(14,29)(15,28)(16,32)(17,31)(18,35)(19,34)(20,33)(21,22)(23,25)(41,42)(43,45)
(46,47)(48,50)(51,82)(52,81)(53,85)(54,84)(55,83)(56,72)(57,71)(58,75)(59,74)
(60,73)(61,77)(62,76)(63,80)(64,79)(65,78)(66,67)(68,70)(86,87)(88,90);;
s2 := (16,41)(17,42)(18,43)(19,44)(20,45)(21,31)(22,32)(23,33)(24,34)(25,35)
(26,36)(27,37)(28,38)(29,39)(30,40)(61,86)(62,87)(63,88)(64,89)(65,90)(66,76)
(67,77)(68,78)(69,79)(70,80)(71,81)(72,82)(73,83)(74,84)(75,85);;
s3 := ( 1,66)( 2,67)( 3,68)( 4,69)( 5,70)( 6,61)( 7,62)( 8,63)( 9,64)(10,65)
(11,71)(12,72)(13,73)(14,74)(15,75)(16,51)(17,52)(18,53)(19,54)(20,55)(21,46)
(22,47)(23,48)(24,49)(25,50)(26,56)(27,57)(28,58)(29,59)(30,60)(31,81)(32,82)
(33,83)(34,84)(35,85)(36,76)(37,77)(38,78)(39,79)(40,80)(41,86)(42,87)(43,88)
(44,89)(45,90);;
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, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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(90)!( 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)(42,45)(43,44)(47,50)(48,49)
(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)
(78,79)(82,85)(83,84)(87,90)(88,89);
s1 := Sym(90)!( 1, 2)( 3, 5)( 6,37)( 7,36)( 8,40)( 9,39)(10,38)(11,27)(12,26)
(13,30)(14,29)(15,28)(16,32)(17,31)(18,35)(19,34)(20,33)(21,22)(23,25)(41,42)
(43,45)(46,47)(48,50)(51,82)(52,81)(53,85)(54,84)(55,83)(56,72)(57,71)(58,75)
(59,74)(60,73)(61,77)(62,76)(63,80)(64,79)(65,78)(66,67)(68,70)(86,87)(88,90);
s2 := Sym(90)!(16,41)(17,42)(18,43)(19,44)(20,45)(21,31)(22,32)(23,33)(24,34)
(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,86)(62,87)(63,88)(64,89)(65,90)
(66,76)(67,77)(68,78)(69,79)(70,80)(71,81)(72,82)(73,83)(74,84)(75,85);
s3 := Sym(90)!( 1,66)( 2,67)( 3,68)( 4,69)( 5,70)( 6,61)( 7,62)( 8,63)( 9,64)
(10,65)(11,71)(12,72)(13,73)(14,74)(15,75)(16,51)(17,52)(18,53)(19,54)(20,55)
(21,46)(22,47)(23,48)(24,49)(25,50)(26,56)(27,57)(28,58)(29,59)(30,60)(31,81)
(32,82)(33,83)(34,84)(35,85)(36,76)(37,77)(38,78)(39,79)(40,80)(41,86)(42,87)
(43,88)(44,89)(45,90);
poly := sub<Sym(90)|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,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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