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# Polytope of Type {3,2,3,4}

Atlas Canonical Name : {3,2,3,4}*144
if this polytope has a name.
Group : SmallGroup(144,183)
Rank : 5
Schlafli Type : {3,2,3,4}
Number of vertices, edges, etc : 3, 3, 3, 6, 4
Order of s0s1s2s3s4 : 3
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,2,3,4,2} of size 288
{3,2,3,4,4} of size 1152
Vertex Figure Of :
{2,3,2,3,4} of size 288
{3,3,2,3,4} of size 576
{4,3,2,3,4} of size 576
{6,3,2,3,4} of size 864
{4,3,2,3,4} of size 1152
{6,3,2,3,4} of size 1152
{5,3,2,3,4} of size 1440
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,2,3,4}*288, {3,2,6,4}*288b, {3,2,6,4}*288c, {6,2,3,4}*288
3-fold covers : {9,2,3,4}*432, {3,2,9,4}*432, {3,6,3,4}*432
4-fold covers : {3,2,12,4}*576b, {3,2,12,4}*576c, {12,2,3,4}*576, {3,2,3,8}*576, {3,2,6,4}*576, {6,2,3,4}*576, {6,2,6,4}*576b, {6,2,6,4}*576c
5-fold covers : {3,2,15,4}*720, {15,2,3,4}*720
6-fold covers : {9,2,3,4}*864, {9,2,6,4}*864b, {9,2,6,4}*864c, {18,2,3,4}*864, {3,2,9,4}*864, {3,2,18,4}*864b, {3,2,18,4}*864c, {6,2,9,4}*864, {3,6,3,4}*864, {3,6,6,4}*864b, {3,6,6,4}*864c, {6,6,3,4}*864a, {3,6,6,4}*864e, {3,2,3,12}*864, {3,2,6,12}*864d, {6,6,3,4}*864b
7-fold covers : {3,2,21,4}*1008, {21,2,3,4}*1008
8-fold covers : {3,2,6,4}*1152a, {3,2,3,8}*1152, {3,2,6,8}*1152a, {3,2,24,4}*1152c, {3,2,24,4}*1152d, {24,2,3,4}*1152, {3,2,12,4}*1152b, {6,2,12,4}*1152b, {6,2,12,4}*1152c, {12,2,3,4}*1152, {12,2,6,4}*1152b, {12,2,6,4}*1152c, {3,2,6,4}*1152b, {3,2,12,4}*1152c, {6,4,6,4}*1152b, {3,2,6,8}*1152b, {6,2,3,8}*1152, {3,2,6,8}*1152c, {3,4,6,4}*1152b, {6,2,6,4}*1152, {6,4,3,4}*1152
9-fold covers : {27,2,3,4}*1296, {3,2,27,4}*1296, {9,2,9,4}*1296, {9,6,3,4}*1296, {3,6,3,4}*1296a, {3,6,9,4}*1296, {3,6,3,4}*1296b
10-fold covers : {3,2,6,20}*1440b, {3,2,15,4}*1440, {3,2,30,4}*1440b, {3,2,30,4}*1440c, {6,2,15,4}*1440, {15,2,3,4}*1440, {15,2,6,4}*1440b, {15,2,6,4}*1440c, {30,2,3,4}*1440
11-fold covers : {3,2,33,4}*1584, {33,2,3,4}*1584
12-fold covers : {9,2,12,4}*1728b, {9,2,12,4}*1728c, {36,2,3,4}*1728, {9,2,3,8}*1728, {3,2,36,4}*1728b, {3,2,36,4}*1728c, {12,2,9,4}*1728, {3,6,12,4}*1728b, {3,6,12,4}*1728c, {12,6,3,4}*1728a, {3,2,9,8}*1728, {3,6,3,8}*1728, {9,2,6,4}*1728, {18,2,3,4}*1728, {18,2,6,4}*1728b, {18,2,6,4}*1728c, {3,2,18,4}*1728, {6,2,9,4}*1728, {6,2,18,4}*1728b, {6,2,18,4}*1728c, {3,6,6,4}*1728a, {6,6,3,4}*1728a, {6,6,6,4}*1728b, {6,6,6,4}*1728c, {3,2,3,24}*1728, {3,6,12,4}*1728e, {3,6,12,4}*1728f, {12,6,3,4}*1728b, {3,6,6,4}*1728b, {6,6,3,4}*1728b, {6,6,6,4}*1728j, {6,6,6,4}*1728k, {6,6,6,4}*1728l, {6,6,6,4}*1728n, {3,2,6,12}*1728a, {3,2,6,12}*1728b, {6,2,3,12}*1728, {6,2,6,12}*1728d
13-fold covers : {3,2,39,4}*1872, {39,2,3,4}*1872
Permutation Representation (GAP) :
```s0 := (2,3);;
s1 := (1,2);;
s2 := (6,7);;
s3 := (5,6);;
s4 := (4,5)(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,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4, s2*s4*s3*s2*s4*s3*s2*s4*s3 ];;
poly := F / rels;;

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

```

to this polytope