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# Polytope of Type {18,3}

Atlas Canonical Name : {18,3}*972b
Also Known As : {18,3}6if this polytope has another name.
Group : SmallGroup(972,115)
Rank : 3
Schlafli Type : {18,3}
Number of vertices, edges, etc : 162, 243, 27
Order of s0s1s2 : 6
Order of s0s1s2s1 : 18
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{18,3,2} of size 1944
Vertex Figure Of :
{2,18,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {18,3}*324
9-fold quotients : {6,3}*108
27-fold quotients : {6,3}*36
81-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {18,6}*1944k
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(20,21)(22,26)
(23,25)(24,27);;
s1 := (10,27)(11,25)(12,26)(13,19)(14,20)(15,21)(16,24)(17,22)(18,23);;
s2 := ( 1,10)( 2,11)( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18);;
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,
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*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*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(27)!( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(20,21)
(22,26)(23,25)(24,27);
s1 := Sym(27)!(10,27)(11,25)(12,26)(13,19)(14,20)(15,21)(16,24)(17,22)(18,23);
s2 := Sym(27)!( 1,10)( 2,11)( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18);
poly := sub<Sym(27)|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, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*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*s0*s1*s0*s1 >;

```
References : None.
to this polytope