# Polytope of Type {2,18,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,2}*144
if this polytope has a name.
Group : SmallGroup(144,112)
Rank : 4
Schlafli Type : {2,18,2}
Number of vertices, edges, etc : 2, 18, 18, 2
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,18,2,2} of size 288
{2,18,2,3} of size 432
{2,18,2,4} of size 576
{2,18,2,5} of size 720
{2,18,2,6} of size 864
{2,18,2,7} of size 1008
{2,18,2,8} of size 1152
{2,18,2,9} of size 1296
{2,18,2,10} of size 1440
{2,18,2,11} of size 1584
{2,18,2,12} of size 1728
{2,18,2,13} of size 1872
Vertex Figure Of :
{2,2,18,2} of size 288
{3,2,18,2} of size 432
{4,2,18,2} of size 576
{5,2,18,2} of size 720
{6,2,18,2} of size 864
{7,2,18,2} of size 1008
{8,2,18,2} of size 1152
{9,2,18,2} of size 1296
{10,2,18,2} of size 1440
{11,2,18,2} of size 1584
{12,2,18,2} of size 1728
{13,2,18,2} of size 1872
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,9,2}*72
3-fold quotients : {2,6,2}*48
6-fold quotients : {2,3,2}*24
9-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,36,2}*288, {2,18,4}*288a, {4,18,2}*288a
3-fold covers : {2,54,2}*432, {2,18,6}*432a, {2,18,6}*432b, {6,18,2}*432a, {6,18,2}*432b
4-fold covers : {2,36,4}*576a, {4,36,2}*576a, {4,18,4}*576a, {2,72,2}*576, {2,18,8}*576, {8,18,2}*576, {2,18,4}*576, {4,18,2}*576
5-fold covers : {2,18,10}*720, {10,18,2}*720, {2,90,2}*720
6-fold covers : {2,108,2}*864, {2,54,4}*864a, {4,54,2}*864a, {2,36,6}*864a, {2,36,6}*864b, {6,36,2}*864a, {6,36,2}*864b, {2,18,12}*864a, {12,18,2}*864a, {4,18,6}*864a, {4,18,6}*864b, {6,18,4}*864a, {6,18,4}*864b, {2,18,12}*864b, {12,18,2}*864b
7-fold covers : {2,18,14}*1008, {14,18,2}*1008, {2,126,2}*1008
8-fold covers : {4,36,4}*1152a, {2,36,8}*1152a, {8,36,2}*1152a, {2,72,4}*1152a, {4,72,2}*1152a, {2,36,8}*1152b, {8,36,2}*1152b, {2,72,4}*1152b, {4,72,2}*1152b, {2,36,4}*1152a, {4,36,2}*1152a, {4,18,8}*1152a, {8,18,4}*1152a, {2,18,16}*1152, {16,18,2}*1152, {2,144,2}*1152, {2,36,4}*1152b, {4,36,2}*1152b, {4,18,4}*1152a, {4,18,4}*1152b, {2,18,4}*1152b, {2,36,4}*1152c, {4,18,2}*1152b, {4,36,2}*1152c, {2,18,8}*1152b, {8,18,2}*1152b, {2,18,8}*1152c, {8,18,2}*1152c
9-fold covers : {2,162,2}*1296, {2,18,18}*1296a, {2,18,18}*1296c, {18,18,2}*1296a, {18,18,2}*1296b, {2,18,6}*1296a, {2,18,6}*1296b, {6,18,2}*1296a, {6,18,2}*1296b, {2,54,6}*1296a, {2,54,6}*1296b, {6,54,2}*1296a, {6,54,2}*1296b, {6,18,6}*1296a, {6,18,6}*1296b, {6,18,6}*1296c, {6,18,6}*1296d, {2,18,6}*1296i, {6,18,2}*1296i
10-fold covers : {2,36,10}*1440, {10,36,2}*1440, {2,18,20}*1440a, {20,18,2}*1440a, {4,18,10}*1440a, {10,18,4}*1440a, {2,180,2}*1440, {2,90,4}*1440a, {4,90,2}*1440a
11-fold covers : {2,18,22}*1584, {22,18,2}*1584, {2,198,2}*1584
12-fold covers : {2,108,4}*1728a, {4,108,2}*1728a, {4,54,4}*1728a, {2,216,2}*1728, {2,54,8}*1728, {8,54,2}*1728, {4,18,12}*1728a, {12,18,4}*1728a, {4,36,6}*1728a, {4,36,6}*1728b, {6,36,4}*1728a, {6,36,4}*1728b, {2,72,6}*1728a, {2,72,6}*1728b, {6,72,2}*1728a, {6,72,2}*1728b, {2,18,24}*1728a, {24,18,2}*1728a, {6,18,8}*1728a, {6,18,8}*1728b, {8,18,6}*1728a, {8,18,6}*1728b, {2,36,12}*1728a, {2,36,12}*1728b, {12,36,2}*1728a, {12,36,2}*1728b, {2,18,24}*1728b, {24,18,2}*1728b, {4,18,12}*1728b, {12,18,4}*1728b, {2,54,4}*1728, {4,54,2}*1728, {2,18,6}*1728, {2,36,6}*1728, {6,18,2}*1728, {6,36,2}*1728, {4,18,6}*1728a, {4,18,6}*1728b, {6,18,4}*1728a, {6,18,4}*1728b, {2,18,12}*1728a, {2,18,12}*1728b, {12,18,2}*1728a, {12,18,2}*1728b
13-fold covers : {2,18,26}*1872, {26,18,2}*1872, {2,234,2}*1872
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20);;
s2 := ( 3, 7)( 4, 5)( 6,11)( 8, 9)(10,15)(12,13)(14,19)(16,17)(18,20);;
s3 := (21,22);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(22)!(1,2);
s1 := Sym(22)!( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20);
s2 := Sym(22)!( 3, 7)( 4, 5)( 6,11)( 8, 9)(10,15)(12,13)(14,19)(16,17)(18,20);
s3 := Sym(22)!(21,22);
poly := sub<Sym(22)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```

to this polytope