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# Polytope of Type {4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12}*432b
if this polytope has a name.
Group : SmallGroup(432,741)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 18, 108, 54
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{4,12,2} of size 864
{4,12,4} of size 1728
{4,12,4} of size 1728
Vertex Figure Of :
{2,4,12} of size 864
{4,4,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {4,4}*144
6-fold quotients : {4,4}*72
18-fold quotients : {2,6}*24
36-fold quotients : {2,3}*12
54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12}*864d
3-fold covers : {4,36}*1296, {4,12}*1296, {12,12}*1296a, {12,12}*1296b, {12,12}*1296f, {12,12}*1296g
4-fold covers : {4,24}*1728f, {8,12}*1728e, {4,24}*1728h, {8,12}*1728f, {4,12}*1728d, {4,12}*1728e
Permutation Representation (GAP) :
```s0 := (4,7)(5,8)(6,9);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17);;
s2 := ( 1, 2)( 4, 5)( 7, 8)(10,17)(11,16)(12,18)(13,14);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(18)!(4,7)(5,8)(6,9);
s1 := Sym(18)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17);
s2 := Sym(18)!( 1, 2)( 4, 5)( 7, 8)(10,17)(11,16)(12,18)(13,14);
poly := sub<Sym(18)|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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

```
References : None.
to this polytope