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# Polytope of Type {6,6,10}

Atlas Canonical Name : {6,6,10}*720b
if this polytope has a name.
Group : SmallGroup(720,813)
Rank : 4
Schlafli Type : {6,6,10}
Number of vertices, edges, etc : 6, 18, 30, 10
Order of s0s1s2s3 : 30
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,6,10,2} of size 1440
Vertex Figure Of :
{2,6,6,10} of size 1440
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,6,10}*240
5-fold quotients : {6,6,2}*144b
9-fold quotients : {2,2,10}*80
10-fold quotients : {6,3,2}*72
15-fold quotients : {2,6,2}*48
18-fold quotients : {2,2,5}*40
30-fold quotients : {2,3,2}*24
45-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,12,10}*1440b, {6,6,20}*1440b, {12,6,10}*1440c
Permutation Representation (GAP) :
```s0 := ( 6,11)( 7,12)( 8,13)( 9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30)
(36,41)(37,42)(38,43)(39,44)(40,45)(51,56)(52,57)(53,58)(54,59)(55,60)(66,71)
(67,72)(68,73)(69,74)(70,75)(81,86)(82,87)(83,88)(84,89)(85,90);;
s1 := ( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(16,36)(17,37)(18,38)(19,39)(20,40)
(21,31)(22,32)(23,33)(24,34)(25,35)(26,41)(27,42)(28,43)(29,44)(30,45)(46,51)
(47,52)(48,53)(49,54)(50,55)(61,81)(62,82)(63,83)(64,84)(65,85)(66,76)(67,77)
(68,78)(69,79)(70,80)(71,86)(72,87)(73,88)(74,89)(75,90);;
s2 := ( 1,16)( 2,20)( 3,19)( 4,18)( 5,17)( 6,26)( 7,30)( 8,29)( 9,28)(10,27)
(11,21)(12,25)(13,24)(14,23)(15,22)(32,35)(33,34)(36,41)(37,45)(38,44)(39,43)
(40,42)(46,61)(47,65)(48,64)(49,63)(50,62)(51,71)(52,75)(53,74)(54,73)(55,72)
(56,66)(57,70)(58,69)(59,68)(60,67)(77,80)(78,79)(81,86)(82,90)(83,89)(84,88)
(85,87);;
s3 := ( 1,47)( 2,46)( 3,50)( 4,49)( 5,48)( 6,52)( 7,51)( 8,55)( 9,54)(10,53)
(11,57)(12,56)(13,60)(14,59)(15,58)(16,62)(17,61)(18,65)(19,64)(20,63)(21,67)
(22,66)(23,70)(24,69)(25,68)(26,72)(27,71)(28,75)(29,74)(30,73)(31,77)(32,76)
(33,80)(34,79)(35,78)(36,82)(37,81)(38,85)(39,84)(40,83)(41,87)(42,86)(43,90)
(44,89)(45,88);;
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*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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(90)!( 6,11)( 7,12)( 8,13)( 9,14)(10,15)(21,26)(22,27)(23,28)(24,29)
(25,30)(36,41)(37,42)(38,43)(39,44)(40,45)(51,56)(52,57)(53,58)(54,59)(55,60)
(66,71)(67,72)(68,73)(69,74)(70,75)(81,86)(82,87)(83,88)(84,89)(85,90);
s1 := Sym(90)!( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(16,36)(17,37)(18,38)(19,39)
(20,40)(21,31)(22,32)(23,33)(24,34)(25,35)(26,41)(27,42)(28,43)(29,44)(30,45)
(46,51)(47,52)(48,53)(49,54)(50,55)(61,81)(62,82)(63,83)(64,84)(65,85)(66,76)
(67,77)(68,78)(69,79)(70,80)(71,86)(72,87)(73,88)(74,89)(75,90);
s2 := Sym(90)!( 1,16)( 2,20)( 3,19)( 4,18)( 5,17)( 6,26)( 7,30)( 8,29)( 9,28)
(10,27)(11,21)(12,25)(13,24)(14,23)(15,22)(32,35)(33,34)(36,41)(37,45)(38,44)
(39,43)(40,42)(46,61)(47,65)(48,64)(49,63)(50,62)(51,71)(52,75)(53,74)(54,73)
(55,72)(56,66)(57,70)(58,69)(59,68)(60,67)(77,80)(78,79)(81,86)(82,90)(83,89)
(84,88)(85,87);
s3 := Sym(90)!( 1,47)( 2,46)( 3,50)( 4,49)( 5,48)( 6,52)( 7,51)( 8,55)( 9,54)
(10,53)(11,57)(12,56)(13,60)(14,59)(15,58)(16,62)(17,61)(18,65)(19,64)(20,63)
(21,67)(22,66)(23,70)(24,69)(25,68)(26,72)(27,71)(28,75)(29,74)(30,73)(31,77)
(32,76)(33,80)(34,79)(35,78)(36,82)(37,81)(38,85)(39,84)(40,83)(41,87)(42,86)
(43,90)(44,89)(45,88);
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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