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# Polytope of Type {15,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {15,2}*60
if this polytope has a name.
Group : SmallGroup(60,12)
Rank : 3
Schlafli Type : {15,2}
Number of vertices, edges, etc : 15, 15, 2
Order of s0s1s2 : 30
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{15,2,2} of size 120
{15,2,3} of size 180
{15,2,4} of size 240
{15,2,5} of size 300
{15,2,6} of size 360
{15,2,7} of size 420
{15,2,8} of size 480
{15,2,9} of size 540
{15,2,10} of size 600
{15,2,11} of size 660
{15,2,12} of size 720
{15,2,13} of size 780
{15,2,14} of size 840
{15,2,15} of size 900
{15,2,16} of size 960
{15,2,17} of size 1020
{15,2,18} of size 1080
{15,2,19} of size 1140
{15,2,20} of size 1200
{15,2,21} of size 1260
{15,2,22} of size 1320
{15,2,23} of size 1380
{15,2,24} of size 1440
{15,2,25} of size 1500
{15,2,26} of size 1560
{15,2,27} of size 1620
{15,2,28} of size 1680
{15,2,29} of size 1740
{15,2,30} of size 1800
{15,2,31} of size 1860
{15,2,32} of size 1920
{15,2,33} of size 1980
Vertex Figure Of :
{2,15,2} of size 120
{4,15,2} of size 240
{6,15,2} of size 360
{6,15,2} of size 480
{4,15,2} of size 480
{10,15,2} of size 600
{3,15,2} of size 720
{6,15,2} of size 720
{10,15,2} of size 720
{12,15,2} of size 960
{8,15,2} of size 960
{4,15,2} of size 960
{6,15,2} of size 1080
{5,15,2} of size 1200
{10,15,2} of size 1200
{4,15,2} of size 1440
{6,15,2} of size 1440
{6,15,2} of size 1440
{6,15,2} of size 1440
{6,15,2} of size 1440
{10,15,2} of size 1440
{12,15,2} of size 1440
{6,15,2} of size 1440
{30,15,2} of size 1800
{6,15,2} of size 1920
{8,15,2} of size 1920
{8,15,2} of size 1920
{8,15,2} of size 1920
{10,15,2} of size 1920
{4,15,2} of size 1920
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {5,2}*20
5-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {30,2}*120
3-fold covers : {45,2}*180, {15,6}*180
4-fold covers : {60,2}*240, {30,4}*240a, {15,4}*240
5-fold covers : {75,2}*300, {15,10}*300
6-fold covers : {90,2}*360, {30,6}*360b, {30,6}*360c
7-fold covers : {105,2}*420
8-fold covers : {60,4}*480a, {120,2}*480, {30,8}*480, {15,8}*480, {30,4}*480
9-fold covers : {135,2}*540, {45,6}*540, {15,6}*540
10-fold covers : {150,2}*600, {30,10}*600b, {30,10}*600c
11-fold covers : {165,2}*660
12-fold covers : {180,2}*720, {90,4}*720a, {45,4}*720, {30,12}*720b, {60,6}*720b, {60,6}*720c, {30,12}*720c, {15,12}*720, {15,6}*720e
13-fold covers : {195,2}*780
14-fold covers : {30,14}*840, {210,2}*840
15-fold covers : {225,2}*900, {75,6}*900, {45,10}*900, {15,30}*900
16-fold covers : {120,4}*960a, {60,4}*960a, {120,4}*960b, {60,8}*960a, {60,8}*960b, {240,2}*960, {30,16}*960, {15,8}*960a, {60,4}*960b, {30,4}*960b, {60,4}*960c, {30,8}*960b, {30,8}*960c, {15,4}*960
17-fold covers : {255,2}*1020
18-fold covers : {270,2}*1080, {90,6}*1080a, {90,6}*1080b, {30,18}*1080b, {30,6}*1080b, {30,6}*1080c, {30,6}*1080d
19-fold covers : {285,2}*1140
20-fold covers : {300,2}*1200, {150,4}*1200a, {75,4}*1200, {30,20}*1200b, {60,10}*1200b, {60,10}*1200c, {30,20}*1200c, {15,20}*1200
21-fold covers : {315,2}*1260, {105,6}*1260
22-fold covers : {30,22}*1320, {330,2}*1320
23-fold covers : {345,2}*1380
24-fold covers : {180,4}*1440a, {360,2}*1440, {90,8}*1440, {45,8}*1440, {30,24}*1440b, {120,6}*1440b, {120,6}*1440c, {60,12}*1440b, {60,12}*1440c, {30,24}*1440c, {90,4}*1440, {15,24}*1440, {15,12}*1440c, {30,12}*1440a, {30,12}*1440b, {30,6}*1440h, {60,6}*1440d
25-fold covers : {375,2}*1500, {75,10}*1500, {15,10}*1500e, {15,10}*1500g
26-fold covers : {30,26}*1560, {390,2}*1560
27-fold covers : {405,2}*1620, {45,18}*1620, {45,6}*1620a, {135,6}*1620, {45,6}*1620b, {45,6}*1620c, {45,6}*1620d, {15,6}*1620, {15,18}*1620
28-fold covers : {60,14}*1680, {30,28}*1680a, {420,2}*1680, {210,4}*1680a, {105,4}*1680
29-fold covers : {435,2}*1740
30-fold covers : {450,2}*1800, {150,6}*1800b, {150,6}*1800c, {90,10}*1800b, {90,10}*1800c, {30,30}*1800a, {30,30}*1800f, {30,30}*1800g, {30,30}*1800i
31-fold covers : {465,2}*1860
32-fold covers : {60,8}*1920a, {120,4}*1920a, {120,8}*1920a, {120,8}*1920b, {120,8}*1920c, {120,8}*1920d, {60,16}*1920a, {240,4}*1920a, {60,16}*1920b, {240,4}*1920b, {60,4}*1920a, {120,4}*1920b, {60,8}*1920b, {30,32}*1920, {480,2}*1920, {15,8}*1920a, {30,8}*1920a, {60,4}*1920d, {60,8}*1920e, {60,8}*1920f, {30,4}*1920a, {30,8}*1920d, {30,8}*1920e, {30,8}*1920f, {60,8}*1920g, {60,8}*1920h, {120,4}*1920c, {120,4}*1920d, {30,8}*1920g, {60,4}*1920e, {120,4}*1920e, {30,4}*1920b, {120,4}*1920f, {15,8}*1920b, {15,4}*1920a, {15,8}*1920c, {30,4}*1920c, {30,4}*1920d
33-fold covers : {495,2}*1980, {165,6}*1980
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15);;
s1 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14);;
s2 := (16,17);;
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, s1*s2*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*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(17)!( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15);
s1 := Sym(17)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14);
s2 := Sym(17)!(16,17);
poly := sub<Sym(17)|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, s1*s2*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*s0*s1*s0*s1*s0*s1 >;

```

to this polytope