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# Polytope of Type {6,2,2,3}

Atlas Canonical Name : {6,2,2,3}*144
if this polytope has a name.
Group : SmallGroup(144,192)
Rank : 5
Schlafli Type : {6,2,2,3}
Number of vertices, edges, etc : 6, 6, 2, 3, 3
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,2,2,3,2} of size 288
{6,2,2,3,3} of size 576
{6,2,2,3,4} of size 576
{6,2,2,3,6} of size 864
{6,2,2,3,4} of size 1152
{6,2,2,3,6} of size 1152
{6,2,2,3,5} of size 1440
Vertex Figure Of :
{2,6,2,2,3} of size 288
{3,6,2,2,3} of size 432
{4,6,2,2,3} of size 576
{3,6,2,2,3} of size 576
{4,6,2,2,3} of size 576
{4,6,2,2,3} of size 576
{4,6,2,2,3} of size 864
{6,6,2,2,3} of size 864
{6,6,2,2,3} of size 864
{6,6,2,2,3} of size 864
{8,6,2,2,3} of size 1152
{4,6,2,2,3} of size 1152
{6,6,2,2,3} of size 1152
{9,6,2,2,3} of size 1296
{3,6,2,2,3} of size 1296
{6,6,2,2,3} of size 1296
{4,6,2,2,3} of size 1440
{5,6,2,2,3} of size 1440
{6,6,2,2,3} of size 1440
{5,6,2,2,3} of size 1440
{5,6,2,2,3} of size 1440
{10,6,2,2,3} of size 1440
{12,6,2,2,3} of size 1728
{12,6,2,2,3} of size 1728
{12,6,2,2,3} of size 1728
{3,6,2,2,3} of size 1728
{12,6,2,2,3} of size 1728
{4,6,2,2,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,2,2,3}*72
3-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,2,2,3}*288, {6,4,2,3}*288a, {6,2,2,6}*288
3-fold covers : {6,2,2,9}*432, {18,2,2,3}*432, {6,2,6,3}*432, {6,6,2,3}*432a, {6,6,2,3}*432c
4-fold covers : {12,4,2,3}*576a, {24,2,2,3}*576, {6,8,2,3}*576, {6,2,2,12}*576, {12,2,2,6}*576, {6,2,4,6}*576a, {6,4,2,6}*576a, {6,2,4,3}*576, {6,4,2,3}*576
5-fold covers : {6,10,2,3}*720, {6,2,2,15}*720, {30,2,2,3}*720
6-fold covers : {36,2,2,3}*864, {12,2,2,9}*864, {6,4,2,9}*864a, {18,4,2,3}*864a, {6,2,2,18}*864, {18,2,2,6}*864, {6,12,2,3}*864a, {12,2,6,3}*864, {12,6,2,3}*864a, {12,6,2,3}*864b, {6,4,6,3}*864, {6,12,2,3}*864c, {6,2,6,6}*864a, {6,2,6,6}*864b, {6,6,2,6}*864a, {6,6,2,6}*864c
7-fold covers : {6,14,2,3}*1008, {6,2,2,21}*1008, {42,2,2,3}*1008
8-fold covers : {12,8,2,3}*1152a, {24,4,2,3}*1152a, {12,8,2,3}*1152b, {24,4,2,3}*1152b, {12,4,2,3}*1152a, {6,16,2,3}*1152, {48,2,2,3}*1152, {6,4,4,6}*1152, {6,2,4,12}*1152a, {12,4,2,6}*1152a, {6,4,2,12}*1152a, {12,2,4,6}*1152a, {12,2,2,12}*1152, {6,2,8,6}*1152, {6,8,2,6}*1152, {6,2,2,24}*1152, {24,2,2,6}*1152, {12,4,2,3}*1152b, {12,2,4,3}*1152, {6,4,4,3}*1152b, {6,4,2,3}*1152b, {12,4,2,3}*1152c, {6,2,8,3}*1152, {6,8,2,3}*1152b, {6,8,2,3}*1152c, {6,2,4,6}*1152, {6,4,2,6}*1152
9-fold covers : {18,2,2,9}*1296, {6,2,2,27}*1296, {54,2,2,3}*1296, {6,2,6,9}*1296, {6,6,2,9}*1296a, {6,6,2,9}*1296c, {6,18,2,3}*1296a, {18,2,6,3}*1296, {18,6,2,3}*1296a, {18,6,2,3}*1296b, {6,6,6,3}*1296a, {6,2,6,3}*1296, {6,6,2,3}*1296b, {6,6,2,3}*1296c, {6,6,6,3}*1296c, {6,6,2,3}*1296d, {6,6,6,3}*1296e
10-fold covers : {12,10,2,3}*1440, {6,20,2,3}*1440a, {12,2,2,15}*1440, {60,2,2,3}*1440, {6,4,2,15}*1440a, {30,4,2,3}*1440a, {6,2,10,6}*1440, {6,10,2,6}*1440, {6,2,2,30}*1440, {30,2,2,6}*1440
11-fold covers : {6,22,2,3}*1584, {6,2,2,33}*1584, {66,2,2,3}*1584
12-fold covers : {12,4,2,9}*1728a, {36,4,2,3}*1728a, {72,2,2,3}*1728, {24,2,2,9}*1728, {6,8,2,9}*1728, {18,8,2,3}*1728, {12,2,2,18}*1728, {18,2,2,12}*1728, {6,2,2,36}*1728, {36,2,2,6}*1728, {6,2,4,18}*1728a, {6,4,2,18}*1728a, {18,2,4,6}*1728a, {18,4,2,6}*1728a, {6,24,2,3}*1728a, {24,2,6,3}*1728, {24,6,2,3}*1728a, {24,6,2,3}*1728b, {12,12,2,3}*1728a, {12,12,2,3}*1728c, {12,4,6,3}*1728, {6,8,6,3}*1728, {6,24,2,3}*1728c, {6,4,2,9}*1728, {18,2,4,3}*1728, {6,2,4,9}*1728, {18,4,2,3}*1728, {6,2,6,12}*1728a, {6,2,6,12}*1728b, {6,2,12,6}*1728a, {6,6,2,12}*1728a, {6,6,2,12}*1728c, {6,12,2,6}*1728a, {12,2,6,6}*1728a, {12,2,6,6}*1728b, {12,6,2,6}*1728a, {12,6,2,6}*1728b, {6,4,6,6}*1728a, {6,6,4,6}*1728a, {6,4,6,6}*1728c, {6,6,4,6}*1728c, {6,2,12,6}*1728c, {6,12,2,6}*1728c, {6,4,6,3}*1728b, {6,6,4,3}*1728a, {6,6,4,3}*1728c, {6,2,6,3}*1728, {6,2,12,3}*1728, {6,6,2,3}*1728b, {6,12,2,3}*1728a, {6,12,2,3}*1728b, {12,6,2,3}*1728a
13-fold covers : {6,26,2,3}*1872, {6,2,2,39}*1872, {78,2,2,3}*1872
Permutation Representation (GAP) :
```s0 := (3,4)(5,6);;
s1 := (1,5)(2,3)(4,6);;
s2 := (7,8);;
s3 := (10,11);;
s4 := ( 9,10);;
poly := Group([s0,s1,s2,s3,s4]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s3*s4*s3*s4*s3*s4, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(11)!(3,4)(5,6);
s1 := Sym(11)!(1,5)(2,3)(4,6);
s2 := Sym(11)!(7,8);
s3 := Sym(11)!(10,11);
s4 := Sym(11)!( 9,10);
poly := sub<Sym(11)|s0,s1,s2,s3,s4>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s3*s4*s3*s4*s3*s4, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```

to this polytope