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# Polytope of Type {6,6,2}

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

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

```

to this polytope