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# Polytope of Type {15,6}

Atlas Canonical Name : {15,6}*180
if this polytope has a name.
Group : SmallGroup(180,29)
Rank : 3
Schlafli Type : {15,6}
Number of vertices, edges, etc : 15, 45, 6
Order of s0s1s2 : 30
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{15,6,2} of size 360
{15,6,3} of size 540
{15,6,4} of size 720
{15,6,6} of size 1080
{15,6,6} of size 1080
{15,6,8} of size 1440
{15,6,9} of size 1620
{15,6,3} of size 1620
{15,6,10} of size 1800
Vertex Figure Of :
{2,15,6} of size 360
{4,15,6} of size 720
{6,15,6} of size 1080
{4,15,6} of size 1440
{10,15,6} of size 1800
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {15,2}*60
5-fold quotients : {3,6}*36
9-fold quotients : {5,2}*20
15-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {30,6}*360c
3-fold covers : {45,6}*540, {15,6}*540
4-fold covers : {60,6}*720c, {30,12}*720c, {15,12}*720, {15,6}*720e
5-fold covers : {75,6}*900, {15,30}*900
6-fold covers : {90,6}*1080b, {30,6}*1080b, {30,6}*1080d
7-fold covers : {105,6}*1260
8-fold covers : {120,6}*1440c, {60,12}*1440c, {30,24}*1440c, {15,24}*1440, {15,12}*1440c, {30,12}*1440b, {30,6}*1440h
9-fold covers : {45,18}*1620, {45,6}*1620a, {135,6}*1620, {45,6}*1620b, {45,6}*1620c, {45,6}*1620d, {15,6}*1620, {15,18}*1620
10-fold covers : {150,6}*1800c, {30,30}*1800a, {30,30}*1800i
11-fold covers : {165,6}*1980
Permutation Representation (GAP) :
```s0 := ( 2, 5)( 3, 4)( 6,11)( 7,15)( 8,14)( 9,13)(10,12)(16,31)(17,35)(18,34)
(19,33)(20,32)(21,41)(22,45)(23,44)(24,43)(25,42)(26,36)(27,40)(28,39)(29,38)
(30,37);;
s1 := ( 1,22)( 2,21)( 3,25)( 4,24)( 5,23)( 6,17)( 7,16)( 8,20)( 9,19)(10,18)
(11,27)(12,26)(13,30)(14,29)(15,28)(31,37)(32,36)(33,40)(34,39)(35,38)(41,42)
(43,45);;
s2 := (16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)
(26,41)(27,42)(28,43)(29,44)(30,45);;
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*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(45)!( 2, 5)( 3, 4)( 6,11)( 7,15)( 8,14)( 9,13)(10,12)(16,31)(17,35)
(18,34)(19,33)(20,32)(21,41)(22,45)(23,44)(24,43)(25,42)(26,36)(27,40)(28,39)
(29,38)(30,37);
s1 := Sym(45)!( 1,22)( 2,21)( 3,25)( 4,24)( 5,23)( 6,17)( 7,16)( 8,20)( 9,19)
(10,18)(11,27)(12,26)(13,30)(14,29)(15,28)(31,37)(32,36)(33,40)(34,39)(35,38)
(41,42)(43,45);
s2 := Sym(45)!(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)
(25,40)(26,41)(27,42)(28,43)(29,44)(30,45);
poly := sub<Sym(45)|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*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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