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# Polytope of Type {10,15}

Atlas Canonical Name : {10,15}*300
if this polytope has a name.
Group : SmallGroup(300,39)
Rank : 3
Schlafli Type : {10,15}
Number of vertices, edges, etc : 10, 75, 15
Order of s0s1s2 : 30
Order of s0s1s2s1 : 10
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{10,15,2} of size 600
{10,15,4} of size 1200
{10,15,6} of size 1800
Vertex Figure Of :
{2,10,15} of size 600
{4,10,15} of size 1200
{5,10,15} of size 1500
{6,10,15} of size 1800
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {10,5}*100
5-fold quotients : {2,15}*60
15-fold quotients : {2,5}*20
25-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {10,30}*600c
3-fold covers : {10,45}*900, {30,15}*900
4-fold covers : {10,60}*1200c, {20,30}*1200c, {20,15}*1200
5-fold covers : {10,75}*1500, {10,15}*1500e
6-fold covers : {10,90}*1800c, {30,30}*1800d, {30,30}*1800h
Permutation Representation (GAP) :
```s0 := ( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)
(31,46)(32,47)(33,48)(34,49)(35,50)(36,41)(37,42)(38,43)(39,44)(40,45)(56,71)
(57,72)(58,73)(59,74)(60,75)(61,66)(62,67)(63,68)(64,69)(65,70);;
s1 := ( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(11,21)(12,25)(13,24)(14,23)(15,22)
(17,20)(18,19)(26,56)(27,60)(28,59)(29,58)(30,57)(31,51)(32,55)(33,54)(34,53)
(35,52)(36,71)(37,75)(38,74)(39,73)(40,72)(41,66)(42,70)(43,69)(44,68)(45,67)
(46,61)(47,65)(48,64)(49,63)(50,62);;
s2 := ( 1,27)( 2,26)( 3,30)( 4,29)( 5,28)( 6,47)( 7,46)( 8,50)( 9,49)(10,48)
(11,42)(12,41)(13,45)(14,44)(15,43)(16,37)(17,36)(18,40)(19,39)(20,38)(21,32)
(22,31)(23,35)(24,34)(25,33)(51,52)(53,55)(56,72)(57,71)(58,75)(59,74)(60,73)
(61,67)(62,66)(63,70)(64,69)(65,68);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(75)!( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)
(15,20)(31,46)(32,47)(33,48)(34,49)(35,50)(36,41)(37,42)(38,43)(39,44)(40,45)
(56,71)(57,72)(58,73)(59,74)(60,75)(61,66)(62,67)(63,68)(64,69)(65,70);
s1 := Sym(75)!( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(11,21)(12,25)(13,24)(14,23)
(15,22)(17,20)(18,19)(26,56)(27,60)(28,59)(29,58)(30,57)(31,51)(32,55)(33,54)
(34,53)(35,52)(36,71)(37,75)(38,74)(39,73)(40,72)(41,66)(42,70)(43,69)(44,68)
(45,67)(46,61)(47,65)(48,64)(49,63)(50,62);
s2 := Sym(75)!( 1,27)( 2,26)( 3,30)( 4,29)( 5,28)( 6,47)( 7,46)( 8,50)( 9,49)
(10,48)(11,42)(12,41)(13,45)(14,44)(15,43)(16,37)(17,36)(18,40)(19,39)(20,38)
(21,32)(22,31)(23,35)(24,34)(25,33)(51,52)(53,55)(56,72)(57,71)(58,75)(59,74)
(60,73)(61,67)(62,66)(63,70)(64,69)(65,68);
poly := sub<Sym(75)|s0,s1,s2>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope