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

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

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

```

to this polytope