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# Polytope of Type {2,6,6}

Atlas Canonical Name : {2,6,6}*144a
if this polytope has a name.
Group : SmallGroup(144,192)
Rank : 4
Schlafli Type : {2,6,6}
Number of vertices, edges, etc : 2, 6, 18, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,6,6,2} of size 288
{2,6,6,3} of size 432
{2,6,6,4} of size 576
{2,6,6,3} of size 576
{2,6,6,4} of size 576
{2,6,6,6} of size 864
{2,6,6,6} of size 864
{2,6,6,6} of size 864
{2,6,6,8} of size 1152
{2,6,6,4} of size 1152
{2,6,6,6} of size 1152
{2,6,6,9} of size 1296
{2,6,6,3} of size 1296
{2,6,6,5} of size 1440
{2,6,6,5} of size 1440
{2,6,6,10} of size 1440
{2,6,6,12} of size 1728
{2,6,6,12} of size 1728
{2,6,6,12} of size 1728
{2,6,6,3} of size 1728
{2,6,6,4} of size 1728
Vertex Figure Of :
{2,2,6,6} of size 288
{3,2,6,6} of size 432
{4,2,6,6} of size 576
{5,2,6,6} of size 720
{6,2,6,6} of size 864
{7,2,6,6} of size 1008
{8,2,6,6} of size 1152
{9,2,6,6} of size 1296
{10,2,6,6} of size 1440
{11,2,6,6} of size 1584
{12,2,6,6} of size 1728
{13,2,6,6} of size 1872
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,2,6}*48, {2,6,2}*48
6-fold quotients : {2,2,3}*24, {2,3,2}*24
9-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,6,12}*288a, {2,12,6}*288a, {4,6,6}*288a
3-fold covers : {2,6,18}*432a, {2,18,6}*432a, {2,6,6}*432b, {6,6,6}*432b, {6,6,6}*432d, {2,6,6}*432d
4-fold covers : {4,12,6}*576a, {4,6,12}*576a, {2,6,24}*576a, {2,24,6}*576a, {8,6,6}*576a, {2,12,12}*576a, {4,6,6}*576a, {2,6,12}*576a, {2,12,6}*576a
5-fold covers : {10,6,6}*720a, {2,6,30}*720b, {2,30,6}*720b
6-fold covers : {2,6,36}*864a, {2,36,6}*864a, {2,12,18}*864a, {2,18,12}*864a, {2,6,12}*864b, {2,12,6}*864b, {4,6,18}*864a, {4,18,6}*864a, {4,6,6}*864b, {6,6,12}*864b, {6,6,12}*864d, {6,12,6}*864b, {6,12,6}*864c, {12,6,6}*864b, {2,6,12}*864g, {2,12,6}*864g, {4,6,6}*864h, {12,6,6}*864f
7-fold covers : {14,6,6}*1008a, {2,6,42}*1008b, {2,42,6}*1008b
8-fold covers : {4,12,12}*1152b, {8,12,6}*1152b, {4,24,6}*1152c, {2,12,24}*1152a, {2,24,12}*1152a, {8,12,6}*1152e, {4,24,6}*1152f, {2,12,24}*1152d, {2,24,12}*1152d, {4,12,6}*1152b, {2,12,12}*1152a, {8,6,12}*1152b, {4,6,24}*1152b, {16,6,6}*1152a, {2,6,48}*1152b, {2,48,6}*1152b, {4,12,6}*1152e, {2,12,12}*1152d, {2,12,12}*1152f, {4,6,12}*1152a, {2,6,12}*1152b, {2,12,6}*1152b, {4,6,6}*1152d, {4,6,12}*1152b, {4,12,6}*1152g, {4,12,6}*1152h, {2,6,24}*1152c, {2,24,6}*1152c, {8,6,6}*1152b, {2,6,24}*1152e, {2,24,6}*1152e, {8,6,6}*1152d, {2,12,12}*1152j, {2,12,12}*1152k
9-fold covers : {2,18,18}*1296a, {2,6,18}*1296b, {2,18,6}*1296b, {2,6,54}*1296a, {2,54,6}*1296a, {2,6,6}*1296a, {2,6,6}*1296b, {2,6,18}*1296f, {2,18,6}*1296f, {2,6,18}*1296g, {2,18,6}*1296g, {6,6,18}*1296b, {6,6,18}*1296d, {6,18,6}*1296a, {6,18,6}*1296b, {18,6,6}*1296b, {2,6,18}*1296i, {2,18,6}*1296i, {6,6,6}*1296g, {6,6,6}*1296h, {6,6,6}*1296i, {6,6,6}*1296j, {2,6,6}*1296e, {2,6,6}*1296f, {2,6,6}*1296g, {6,6,6}*1296q, {6,6,6}*1296r, {6,6,6}*1296s
10-fold covers : {10,6,12}*1440a, {10,12,6}*1440a, {20,6,6}*1440a, {2,12,30}*1440b, {2,30,12}*1440b, {2,6,60}*1440b, {2,60,6}*1440b, {4,6,30}*1440b, {4,30,6}*1440b
11-fold covers : {22,6,6}*1584a, {2,6,66}*1584b, {2,66,6}*1584b
12-fold covers : {4,6,36}*1728a, {4,18,12}*1728a, {4,12,18}*1728a, {4,36,6}*1728a, {4,6,12}*1728a, {4,12,6}*1728b, {2,6,72}*1728a, {2,72,6}*1728a, {2,18,24}*1728a, {2,24,18}*1728a, {2,6,24}*1728b, {2,24,6}*1728b, {8,6,18}*1728a, {8,18,6}*1728a, {8,6,6}*1728b, {2,12,36}*1728a, {2,36,12}*1728a, {2,12,12}*1728c, {6,6,24}*1728b, {6,6,24}*1728d, {6,24,6}*1728b, {6,24,6}*1728c, {24,6,6}*1728b, {2,6,24}*1728f, {2,24,6}*1728f, {12,6,12}*1728b, {12,6,12}*1728c, {6,12,12}*1728b, {6,12,12}*1728d, {12,12,6}*1728b, {12,12,6}*1728c, {8,6,6}*1728e, {24,6,6}*1728f, {2,12,12}*1728h, {4,12,6}*1728j, {4,6,12}*1728h, {4,6,18}*1728, {2,6,36}*1728, {2,36,6}*1728, {4,18,6}*1728a, {2,12,18}*1728a, {2,18,12}*1728a, {4,6,6}*1728b, {2,6,12}*1728b, {2,12,6}*1728b, {4,6,6}*1728c, {6,6,6}*1728d, {6,6,12}*1728a, {6,6,12}*1728b, {6,12,6}*1728e, {6,12,6}*1728g, {2,6,6}*1728c, {6,12,6}*1728i, {12,6,6}*1728a, {2,6,12}*1728c, {12,6,6}*1728b, {2,12,6}*1728c
13-fold covers : {26,6,6}*1872a, {2,6,78}*1872b, {2,78,6}*1872b
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 7, 8)(11,12)(13,14)(15,16)(17,18)(19,20);;
s2 := ( 3, 7)( 4,11)( 5,15)( 6,13)( 9,19)(10,17)(14,16)(18,20);;
s3 := ( 3, 9)( 4, 5)( 6,10)( 7,17)( 8,18)(11,13)(12,14)(15,19)(16,20);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

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

```

to this polytope