Questions?
See the FAQ
or other info.

# Polytope of Type {12,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,3}*144
if this polytope has a name.
Group : SmallGroup(144,183)
Rank : 3
Schlafli Type : {12,3}
Number of vertices, edges, etc : 24, 36, 6
Order of s0s1s2 : 6
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{12,3,2} of size 288
{12,3,4} of size 576
{12,3,6} of size 864
{12,3,4} of size 1152
Vertex Figure Of :
{2,12,3} of size 288
{4,12,3} of size 576
{6,12,3} of size 864
{6,12,3} of size 864
{4,12,3} of size 1152
{8,12,3} of size 1152
{10,12,3} of size 1440
{12,12,3} of size 1728
{3,12,3} of size 1728
{12,12,3} of size 1728
{4,12,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {4,3}*48
4-fold quotients : {6,3}*36
6-fold quotients : {4,3}*24
12-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,3}*288, {12,6}*288b
3-fold covers : {12,9}*432, {12,3}*432
4-fold covers : {24,3}*576, {12,12}*576g, {12,12}*576i, {24,6}*576b, {24,6}*576d, {12,6}*576f, {12,3}*576
5-fold covers : {12,15}*720
6-fold covers : {24,9}*864, {24,3}*864, {12,18}*864b, {12,6}*864a, {12,6}*864c
7-fold covers : {12,21}*1008
8-fold covers : {24,3}*1152a, {24,6}*1152a, {24,6}*1152c, {24,12}*1152j, {24,12}*1152l, {24,12}*1152m, {24,12}*1152n, {12,6}*1152c, {24,6}*1152f, {12,24}*1152p, {12,24}*1152r, {12,24}*1152s, {12,24}*1152t, {12,12}*1152o, {24,6}*1152k, {24,6}*1152l, {12,12}*1152r, {12,6}*1152f, {12,3}*1152b, {12,6}*1152g, {24,3}*1152b, {24,3}*1152c, {12,6}*1152j
9-fold covers : {12,27}*1296, {36,9}*1296, {36,3}*1296, {12,3}*1296a, {12,9}*1296a, {12,9}*1296b, {12,9}*1296c, {12,9}*1296d
10-fold covers : {24,15}*1440, {60,6}*1440c, {12,30}*1440b
11-fold covers : {12,33}*1584
12-fold covers : {24,9}*1728, {24,3}*1728, {12,36}*1728f, {12,36}*1728g, {12,12}*1728k, {12,12}*1728n, {24,18}*1728b, {24,18}*1728d, {24,6}*1728b, {24,6}*1728d, {12,18}*1728d, {12,6}*1728f, {12,9}*1728, {12,3}*1728, {24,6}*1728f, {24,6}*1728g, {12,12}*1728w, {12,6}*1728i, {12,12}*1728y
13-fold covers : {12,39}*1872
Permutation Representation (GAP) :
```s0 := ( 1, 2)( 3, 4)( 5,10)( 6, 9)( 7,12)( 8,11);;
s1 := ( 1, 5)( 2, 7)( 3, 6)( 4, 8)(10,11);;
s2 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11);;
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*s1*s2,
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope