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

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

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

```

to this polytope