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

Atlas Canonical Name : {2,3,2,3}*72
if this polytope has a name.
Group : SmallGroup(72,46)
Rank : 5
Schlafli Type : {2,3,2,3}
Number of vertices, edges, etc : 2, 3, 3, 3, 3
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,3,2,3,2} of size 144
{2,3,2,3,3} of size 288
{2,3,2,3,4} of size 288
{2,3,2,3,6} of size 432
{2,3,2,3,4} of size 576
{2,3,2,3,6} of size 576
{2,3,2,3,5} of size 720
{2,3,2,3,8} of size 1152
{2,3,2,3,12} of size 1152
{2,3,2,3,6} of size 1296
{2,3,2,3,5} of size 1440
{2,3,2,3,10} of size 1440
{2,3,2,3,10} of size 1440
{2,3,2,3,6} of size 1728
{2,3,2,3,12} of size 1728
Vertex Figure Of :
{2,2,3,2,3} of size 144
{3,2,3,2,3} of size 216
{4,2,3,2,3} of size 288
{5,2,3,2,3} of size 360
{6,2,3,2,3} of size 432
{7,2,3,2,3} of size 504
{8,2,3,2,3} of size 576
{9,2,3,2,3} of size 648
{10,2,3,2,3} of size 720
{11,2,3,2,3} of size 792
{12,2,3,2,3} of size 864
{13,2,3,2,3} of size 936
{14,2,3,2,3} of size 1008
{15,2,3,2,3} of size 1080
{16,2,3,2,3} of size 1152
{17,2,3,2,3} of size 1224
{18,2,3,2,3} of size 1296
{19,2,3,2,3} of size 1368
{20,2,3,2,3} of size 1440
{21,2,3,2,3} of size 1512
{22,2,3,2,3} of size 1584
{23,2,3,2,3} of size 1656
{24,2,3,2,3} of size 1728
{25,2,3,2,3} of size 1800
{26,2,3,2,3} of size 1872
{27,2,3,2,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,3,2,6}*144, {2,6,2,3}*144
3-fold covers : {2,3,2,9}*216, {2,9,2,3}*216, {2,3,6,3}*216, {6,3,2,3}*216
4-fold covers : {2,3,2,12}*288, {2,12,2,3}*288, {4,6,2,3}*288a, {4,3,2,3}*288, {2,6,2,6}*288
5-fold covers : {2,3,2,15}*360, {2,15,2,3}*360
6-fold covers : {2,3,2,18}*432, {2,6,2,9}*432, {2,9,2,6}*432, {2,18,2,3}*432, {2,3,6,6}*432a, {2,6,6,3}*432a, {2,3,6,6}*432b, {2,6,6,3}*432b, {6,3,2,6}*432, {6,6,2,3}*432a, {6,6,2,3}*432b
7-fold covers : {2,3,2,21}*504, {2,21,2,3}*504
8-fold covers : {4,12,2,3}*576a, {2,3,2,24}*576, {2,24,2,3}*576, {8,6,2,3}*576, {8,3,2,3}*576, {2,6,2,12}*576, {2,12,2,6}*576, {2,6,4,6}*576, {4,6,2,6}*576a, {2,3,4,6}*576, {2,6,4,3}*576, {4,3,2,6}*576, {4,6,2,3}*576
9-fold covers : {2,9,2,9}*648, {2,3,6,9}*648, {2,9,6,3}*648, {2,3,2,27}*648, {2,27,2,3}*648, {2,3,6,3}*648a, {2,3,6,3}*648b, {6,3,2,9}*648, {6,9,2,3}*648, {6,3,6,3}*648, {6,3,2,3}*648
10-fold covers : {10,6,2,3}*720, {2,3,2,30}*720, {2,6,2,15}*720, {2,15,2,6}*720, {2,30,2,3}*720
11-fold covers : {2,3,2,33}*792, {2,33,2,3}*792
12-fold covers : {2,3,2,36}*864, {2,36,2,3}*864, {2,9,2,12}*864, {2,12,2,9}*864, {2,3,6,12}*864a, {2,12,6,3}*864a, {4,6,2,9}*864a, {4,18,2,3}*864a, {4,6,6,3}*864a, {4,3,2,9}*864, {4,9,2,3}*864, {4,3,6,3}*864, {2,6,2,18}*864, {2,18,2,6}*864, {2,6,6,6}*864a, {6,3,2,12}*864, {6,12,2,3}*864a, {6,12,2,3}*864b, {12,6,2,3}*864a, {2,3,6,12}*864b, {2,12,6,3}*864b, {12,6,2,3}*864c, {4,6,6,3}*864d, {6,3,2,3}*864, {12,3,2,3}*864, {2,6,6,6}*864b, {2,6,6,6}*864c, {2,6,6,6}*864g, {6,6,2,6}*864a, {6,6,2,6}*864b
13-fold covers : {2,3,2,39}*936, {2,39,2,3}*936
14-fold covers : {14,6,2,3}*1008, {2,3,2,42}*1008, {2,6,2,21}*1008, {2,21,2,6}*1008, {2,42,2,3}*1008
15-fold covers : {2,3,2,45}*1080, {2,45,2,3}*1080, {2,9,2,15}*1080, {2,15,2,9}*1080, {2,3,6,15}*1080, {2,15,6,3}*1080, {6,3,2,15}*1080, {6,15,2,3}*1080
16-fold covers : {8,12,2,3}*1152a, {4,24,2,3}*1152a, {8,12,2,3}*1152b, {4,24,2,3}*1152b, {4,12,2,3}*1152a, {16,6,2,3}*1152, {2,3,2,48}*1152, {2,48,2,3}*1152, {2,6,4,12}*1152, {2,12,4,6}*1152, {4,12,2,6}*1152a, {4,6,4,6}*1152a, {4,6,2,12}*1152a, {2,12,2,12}*1152, {2,6,8,6}*1152, {8,6,2,6}*1152, {2,6,2,24}*1152, {2,24,2,6}*1152, {8,3,2,3}*1152, {4,12,2,3}*1152b, {2,3,4,12}*1152, {2,12,4,3}*1152, {4,3,2,12}*1152, {4,6,4,3}*1152a, {4,6,2,3}*1152b, {4,12,2,3}*1152c, {2,3,8,6}*1152, {2,6,8,3}*1152, {8,3,2,6}*1152, {8,6,2,3}*1152b, {8,6,2,3}*1152c, {2,3,4,3}*1152, {2,6,4,6}*1152a, {2,6,4,6}*1152b, {4,6,2,6}*1152
17-fold covers : {2,3,2,51}*1224, {2,51,2,3}*1224
18-fold covers : {2,9,2,18}*1296, {2,18,2,9}*1296, {2,3,6,18}*1296a, {2,6,6,9}*1296a, {2,9,6,6}*1296a, {2,18,6,3}*1296a, {2,3,2,54}*1296, {2,6,2,27}*1296, {2,27,2,6}*1296, {2,54,2,3}*1296, {2,3,6,6}*1296a, {2,3,6,6}*1296b, {2,6,6,3}*1296a, {2,6,6,3}*1296b, {2,3,6,18}*1296b, {2,6,6,9}*1296b, {2,9,6,6}*1296b, {2,18,6,3}*1296b, {6,3,2,18}*1296, {6,6,2,9}*1296a, {6,6,2,9}*1296b, {6,9,2,6}*1296, {6,18,2,3}*1296a, {6,18,2,3}*1296b, {18,6,2,3}*1296a, {6,3,6,6}*1296a, {2,3,6,6}*1296c, {2,3,6,6}*1296d, {2,3,6,6}*1296e, {6,6,6,3}*1296a, {2,6,6,3}*1296c, {6,6,6,3}*1296b, {2,6,6,3}*1296d, {2,6,6,3}*1296e, {6,3,2,6}*1296, {6,6,2,3}*1296a, {6,6,2,3}*1296b, {6,3,6,6}*1296b, {6,6,6,3}*1296c, {6,6,6,3}*1296d, {6,6,2,3}*1296d
19-fold covers : {2,3,2,57}*1368, {2,57,2,3}*1368
20-fold covers : {10,12,2,3}*1440, {20,6,2,3}*1440a, {2,12,2,15}*1440, {2,15,2,12}*1440, {2,3,2,60}*1440, {2,60,2,3}*1440, {4,6,2,15}*1440a, {4,30,2,3}*1440a, {4,15,2,3}*1440, {4,3,2,15}*1440, {2,6,10,6}*1440, {10,6,2,6}*1440, {2,6,2,30}*1440, {2,30,2,6}*1440
21-fold covers : {2,3,2,63}*1512, {2,63,2,3}*1512, {2,9,2,21}*1512, {2,21,2,9}*1512, {2,3,6,21}*1512, {2,21,6,3}*1512, {6,3,2,21}*1512, {6,21,2,3}*1512
22-fold covers : {22,6,2,3}*1584, {2,3,2,66}*1584, {2,6,2,33}*1584, {2,33,2,6}*1584, {2,66,2,3}*1584
23-fold covers : {2,3,2,69}*1656, {2,69,2,3}*1656
24-fold covers : {4,12,2,9}*1728a, {4,36,2,3}*1728a, {4,12,6,3}*1728a, {2,3,2,72}*1728, {2,72,2,3}*1728, {2,9,2,24}*1728, {2,24,2,9}*1728, {2,3,6,24}*1728a, {2,24,6,3}*1728a, {8,6,2,9}*1728, {8,18,2,3}*1728, {8,6,6,3}*1728a, {8,3,2,9}*1728, {8,9,2,3}*1728, {8,3,6,3}*1728, {2,12,2,18}*1728, {2,18,2,12}*1728, {2,6,2,36}*1728, {2,36,2,6}*1728, {2,6,6,12}*1728a, {2,12,6,6}*1728a, {2,6,4,18}*1728, {2,18,4,6}*1728, {4,6,2,18}*1728a, {4,18,2,6}*1728a, {4,6,6,6}*1728a, {2,6,12,6}*1728a, {6,3,2,24}*1728, {6,24,2,3}*1728a, {6,24,2,3}*1728b, {24,6,2,3}*1728a, {2,3,6,24}*1728b, {2,24,6,3}*1728b, {12,12,2,3}*1728a, {12,12,2,3}*1728b, {24,6,2,3}*1728c, {8,6,6,3}*1728b, {4,12,6,3}*1728d, {2,3,4,18}*1728, {2,18,4,3}*1728, {4,3,2,18}*1728, {4,6,2,9}*1728, {2,6,4,9}*1728, {2,9,4,6}*1728, {4,9,2,6}*1728, {4,18,2,3}*1728, {4,3,6,6}*1728a, {4,6,6,3}*1728a, {2,3,12,6}*1728a, {2,6,12,3}*1728a, {12,3,2,3}*1728, {24,3,2,3}*1728, {2,6,6,12}*1728b, {2,6,6,12}*1728c, {2,6,12,6}*1728b, {2,12,6,6}*1728b, {2,12,6,6}*1728d, {6,6,2,12}*1728a, {6,6,2,12}*1728b, {6,12,2,6}*1728a, {6,12,2,6}*1728b, {12,6,2,6}*1728a, {4,6,6,6}*1728d, {4,6,6,6}*1728f, {6,6,4,6}*1728a, {6,6,4,6}*1728b, {2,6,6,12}*1728e, {2,12,6,6}*1728e, {2,6,12,6}*1728f, {2,6,12,6}*1728g, {12,6,2,6}*1728c, {4,6,6,6}*1728i, {4,3,6,6}*1728b, {4,6,6,3}*1728b, {6,3,4,6}*1728, {6,6,4,3}*1728a, {6,6,4,3}*1728b, {2,3,6,6}*1728, {2,3,12,6}*1728b, {2,6,6,3}*1728, {2,6,12,3}*1728b, {6,3,2,6}*1728, {6,6,2,3}*1728a, {6,12,2,3}*1728a, {12,3,2,6}*1728, {12,6,2,3}*1728a, {12,6,2,3}*1728b
25-fold covers : {2,3,2,75}*1800, {2,75,2,3}*1800, {10,3,2,3}*1800, {10,15,2,3}*1800, {2,15,2,15}*1800
26-fold covers : {26,6,2,3}*1872, {2,3,2,78}*1872, {2,6,2,39}*1872, {2,39,2,6}*1872, {2,78,2,3}*1872
27-fold covers : {2,9,6,9}*1944, {2,3,6,3}*1944, {2,9,2,27}*1944, {2,27,2,9}*1944, {2,3,6,27}*1944, {2,27,6,3}*1944, {2,3,6,9}*1944a, {2,9,6,3}*1944a, {2,3,6,9}*1944b, {2,9,6,3}*1944b, {2,3,2,81}*1944, {2,81,2,3}*1944, {6,9,2,9}*1944, {18,9,2,3}*1944, {6,3,6,9}*1944, {6,9,2,3}*1944a, {6,9,6,3}*1944, {6,3,2,9}*1944, {6,3,6,3}*1944a, {6,3,2,27}*1944, {6,27,2,3}*1944, {6,3,6,3}*1944b, {6,3,6,3}*1944c, {6,9,2,3}*1944b, {6,9,2,3}*1944c, {6,9,2,3}*1944d, {6,3,2,3}*1944, {18,3,2,3}*1944
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (4,5);;
s2 := (3,4);;
s3 := (7,8);;
s4 := (6,7);;
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*s1*s0*s1,
s0*s2*s0*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, s1*s2*s1*s2*s1*s2, s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

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

```

to this polytope