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# Polytope of Type {6,2,6}

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

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

```

to this polytope