# Polytope of Type {12,10}

Atlas Canonical Name : {12,10}*240
Also Known As : {12,10|2}. if this polytope has another name.
Group : SmallGroup(240,136)
Rank : 3
Schlafli Type : {12,10}
Number of vertices, edges, etc : 12, 60, 10
Order of s0s1s2 : 60
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{12,10,2} of size 480
{12,10,4} of size 960
{12,10,5} of size 1200
{12,10,3} of size 1440
{12,10,5} of size 1440
{12,10,6} of size 1440
{12,10,8} of size 1920
Vertex Figure Of :
{2,12,10} of size 480
{4,12,10} of size 960
{4,12,10} of size 960
{4,12,10} of size 960
{3,12,10} of size 960
{6,12,10} of size 1440
{6,12,10} of size 1440
{6,12,10} of size 1440
{3,12,10} of size 1440
{8,12,10} of size 1920
{8,12,10} of size 1920
{4,12,10} of size 1920
{4,12,10} of size 1920
{4,12,10} of size 1920
{6,12,10} of size 1920
{6,12,10} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,10}*120
3-fold quotients : {4,10}*80
5-fold quotients : {12,2}*48
6-fold quotients : {2,10}*40
10-fold quotients : {6,2}*24
12-fold quotients : {2,5}*20
15-fold quotients : {4,2}*16
20-fold quotients : {3,2}*12
30-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,10}*480, {12,20}*480
3-fold covers : {36,10}*720, {12,30}*720a, {12,30}*720b
4-fold covers : {48,10}*960, {12,20}*960a, {24,20}*960a, {12,40}*960a, {24,20}*960b, {12,40}*960b, {12,20}*960b
5-fold covers : {12,50}*1200, {60,10}*1200a, {60,10}*1200b
6-fold covers : {72,10}*1440, {36,20}*1440, {24,30}*1440a, {12,60}*1440a, {24,30}*1440b, {12,60}*1440b
7-fold covers : {84,10}*1680, {12,70}*1680
8-fold covers : {12,40}*1920a, {24,20}*1920a, {24,40}*1920a, {24,40}*1920b, {24,40}*1920c, {24,40}*1920d, {12,80}*1920a, {48,20}*1920a, {12,80}*1920b, {48,20}*1920b, {12,40}*1920b, {24,20}*1920b, {12,20}*1920a, {96,10}*1920, {12,40}*1920e, {12,40}*1920f, {24,20}*1920c, {24,20}*1920d, {12,20}*1920c
Permutation Representation (GAP) :
```s0 := ( 6,11)( 7,12)( 8,13)( 9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30)
(31,46)(32,47)(33,48)(34,49)(35,50)(36,56)(37,57)(38,58)(39,59)(40,60)(41,51)
(42,52)(43,53)(44,54)(45,55);;
s1 := ( 1,36)( 2,40)( 3,39)( 4,38)( 5,37)( 6,31)( 7,35)( 8,34)( 9,33)(10,32)
(11,41)(12,45)(13,44)(14,43)(15,42)(16,51)(17,55)(18,54)(19,53)(20,52)(21,46)
(22,50)(23,49)(24,48)(25,47)(26,56)(27,60)(28,59)(29,58)(30,57);;
s2 := ( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)(23,25)
(26,27)(28,30)(31,32)(33,35)(36,37)(38,40)(41,42)(43,45)(46,47)(48,50)(51,52)
(53,55)(56,57)(58,60);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope