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# Polytope of Type {6,2,3}

Atlas Canonical Name : {6,2,3}*72
if this polytope has a name.
Group : SmallGroup(72,46)
Rank : 4
Schlafli Type : {6,2,3}
Number of vertices, edges, etc : 6, 6, 3, 3
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,2,3,2} of size 144
{6,2,3,3} of size 288
{6,2,3,4} of size 288
{6,2,3,6} of size 432
{6,2,3,4} of size 576
{6,2,3,6} of size 576
{6,2,3,5} of size 720
{6,2,3,8} of size 1152
{6,2,3,12} of size 1152
{6,2,3,6} of size 1296
{6,2,3,5} of size 1440
{6,2,3,10} of size 1440
{6,2,3,10} of size 1440
{6,2,3,6} of size 1728
{6,2,3,12} of size 1728
Vertex Figure Of :
{2,6,2,3} of size 144
{3,6,2,3} of size 216
{4,6,2,3} of size 288
{3,6,2,3} of size 288
{4,6,2,3} of size 288
{4,6,2,3} of size 288
{4,6,2,3} of size 432
{6,6,2,3} of size 432
{6,6,2,3} of size 432
{6,6,2,3} of size 432
{8,6,2,3} of size 576
{4,6,2,3} of size 576
{6,6,2,3} of size 576
{9,6,2,3} of size 648
{3,6,2,3} of size 648
{6,6,2,3} of size 648
{4,6,2,3} of size 720
{5,6,2,3} of size 720
{6,6,2,3} of size 720
{5,6,2,3} of size 720
{5,6,2,3} of size 720
{10,6,2,3} of size 720
{12,6,2,3} of size 864
{12,6,2,3} of size 864
{12,6,2,3} of size 864
{3,6,2,3} of size 864
{12,6,2,3} of size 864
{4,6,2,3} of size 864
{14,6,2,3} of size 1008
{15,6,2,3} of size 1080
{16,6,2,3} of size 1152
{4,6,2,3} of size 1152
{6,6,2,3} of size 1152
{3,6,2,3} of size 1152
{8,6,2,3} of size 1152
{4,6,2,3} of size 1152
{12,6,2,3} of size 1152
{8,6,2,3} of size 1152
{12,6,2,3} of size 1152
{6,6,2,3} of size 1152
{8,6,2,3} of size 1152
{4,6,2,3} of size 1296
{12,6,2,3} of size 1296
{12,6,2,3} of size 1296
{18,6,2,3} of size 1296
{18,6,2,3} of size 1296
{6,6,2,3} of size 1296
{6,6,2,3} of size 1296
{6,6,2,3} of size 1296
{12,6,2,3} of size 1296
{6,6,2,3} of size 1296
{20,6,2,3} of size 1440
{4,6,2,3} of size 1440
{4,6,2,3} of size 1440
{4,6,2,3} of size 1440
{5,6,2,3} of size 1440
{6,6,2,3} of size 1440
{6,6,2,3} of size 1440
{6,6,2,3} of size 1440
{10,6,2,3} of size 1440
{10,6,2,3} of size 1440
{5,6,2,3} of size 1440
{10,6,2,3} of size 1440
{10,6,2,3} of size 1440
{10,6,2,3} of size 1440
{10,6,2,3} of size 1440
{15,6,2,3} of size 1440
{20,6,2,3} of size 1440
{21,6,2,3} of size 1512
{22,6,2,3} of size 1584
{24,6,2,3} of size 1728
{24,6,2,3} of size 1728
{24,6,2,3} of size 1728
{8,6,2,3} of size 1728
{6,6,2,3} of size 1728
{6,6,2,3} of size 1728
{12,6,2,3} of size 1728
{12,6,2,3} of size 1728
{3,6,2,3} of size 1800
{10,6,2,3} of size 1800
{26,6,2,3} of size 1872
{9,6,2,3} of size 1944
{18,6,2,3} of size 1944
{27,6,2,3} of size 1944
{6,6,2,3} of size 1944
{6,6,2,3} of size 1944
{9,6,2,3} of size 1944
{9,6,2,3} of size 1944
{9,6,2,3} of size 1944
{18,6,2,3} of size 1944
{3,6,2,3} of size 1944
{18,6,2,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,2,3}*36
3-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,2,3}*144, {6,2,6}*144
3-fold covers : {6,2,9}*216, {18,2,3}*216, {6,6,3}*216a, {6,6,3}*216b
4-fold covers : {24,2,3}*288, {6,2,12}*288, {12,2,6}*288, {6,4,6}*288, {6,4,3}*288
5-fold covers : {6,2,15}*360, {30,2,3}*360
6-fold covers : {36,2,3}*432, {12,2,9}*432, {12,6,3}*432a, {6,2,18}*432, {18,2,6}*432, {6,6,6}*432a, {12,6,3}*432b, {6,6,6}*432b, {6,6,6}*432c, {6,6,6}*432g
7-fold covers : {6,2,21}*504, {42,2,3}*504
8-fold covers : {48,2,3}*576, {12,2,12}*576, {6,4,12}*576, {12,4,6}*576, {6,2,24}*576, {24,2,6}*576, {6,8,6}*576, {12,4,3}*576, {6,8,3}*576, {6,4,6}*576a, {6,4,6}*576b
9-fold covers : {18,2,9}*648, {6,6,9}*648a, {18,6,3}*648a, {6,2,27}*648, {54,2,3}*648, {6,6,3}*648a, {6,6,3}*648b, {6,6,9}*648b, {18,6,3}*648b, {6,6,3}*648c, {6,6,3}*648d, {6,6,3}*648e
10-fold covers : {12,2,15}*720, {60,2,3}*720, {6,10,6}*720, {6,2,30}*720, {30,2,6}*720
11-fold covers : {6,2,33}*792, {66,2,3}*792
12-fold covers : {72,2,3}*864, {24,2,9}*864, {24,6,3}*864a, {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, {24,6,3}*864b, {18,4,3}*864, {6,4,9}*864, {6,12,3}*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, {6,6,3}*864, {6,12,3}*864b
13-fold covers : {6,2,39}*936, {78,2,3}*936
14-fold covers : {12,2,21}*1008, {84,2,3}*1008, {6,14,6}*1008, {6,2,42}*1008, {42,2,6}*1008
15-fold covers : {6,2,45}*1080, {90,2,3}*1080, {18,2,15}*1080, {30,2,9}*1080, {6,6,15}*1080a, {30,6,3}*1080a, {6,6,15}*1080b, {30,6,3}*1080b
16-fold covers : {96,2,3}*1152, {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, {12,8,3}*1152, {12,4,3}*1152, {6,8,3}*1152, {24,4,3}*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, {6,4,3}*1152b
17-fold covers : {6,2,51}*1224, {102,2,3}*1224
18-fold covers : {36,2,9}*1296, {12,6,9}*1296a, {36,6,3}*1296a, {12,2,27}*1296, {108,2,3}*1296, {12,6,3}*1296a, {12,6,3}*1296b, {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, {36,6,3}*1296b, {12,6,9}*1296b, {12,6,3}*1296c, {12,6,3}*1296d, {12,6,3}*1296e, {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, {12,6,3}*1296f, {6,6,6}*1296q, {6,6,6}*1296s
19-fold covers : {6,2,57}*1368, {114,2,3}*1368
20-fold covers : {24,2,15}*1440, {120,2,3}*1440, {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, {6,4,15}*1440, {30,4,3}*1440
21-fold covers : {6,2,63}*1512, {126,2,3}*1512, {18,2,21}*1512, {42,2,9}*1512, {6,6,21}*1512a, {42,6,3}*1512a, {6,6,21}*1512b, {42,6,3}*1512b
22-fold covers : {12,2,33}*1584, {132,2,3}*1584, {6,22,6}*1584, {6,2,66}*1584, {66,2,6}*1584
23-fold covers : {6,2,69}*1656, {138,2,3}*1656
24-fold covers : {144,2,3}*1728, {48,2,9}*1728, {48,6,3}*1728a, {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, {48,6,3}*1728b, {36,4,3}*1728, {18,8,3}*1728, {12,4,9}*1728, {12,12,3}*1728a, {6,8,9}*1728, {6,24,3}*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, {6,12,3}*1728, {6,24,3}*1728b, {12,6,3}*1728, {12,12,3}*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
25-fold covers : {6,2,75}*1800, {150,2,3}*1800, {6,10,3}*1800, {6,10,15}*1800, {30,2,15}*1800
26-fold covers : {12,2,39}*1872, {156,2,3}*1872, {6,26,6}*1872, {6,2,78}*1872, {78,2,6}*1872
27-fold covers : {18,6,9}*1944a, {6,6,3}*1944a, {18,2,27}*1944, {54,2,9}*1944, {6,6,27}*1944a, {54,6,3}*1944a, {6,6,9}*1944a, {18,6,3}*1944a, {6,6,9}*1944b, {18,6,3}*1944b, {6,2,81}*1944, {162,2,3}*1944, {6,18,9}*1944, {18,6,9}*1944b, {6,6,9}*1944c, {6,6,9}*1944d, {18,6,3}*1944c, {18,6,3}*1944d, {6,6,9}*1944e, {18,6,3}*1944e, {6,6,3}*1944b, {6,6,3}*1944c, {6,6,3}*1944d, {6,6,27}*1944b, {54,6,3}*1944b, {6,6,3}*1944e, {6,6,3}*1944f, {6,6,3}*1944g, {6,6,9}*1944f, {6,6,9}*1944g, {6,6,9}*1944h, {6,6,3}*1944h, {6,18,3}*1944
Permutation Representation (GAP) :
```s0 := (3,4)(5,6);;
s1 := (1,5)(2,3)(4,6);;
s2 := (8,9);;
s3 := (7,8);;
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,
s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```

to this polytope