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

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

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

```

to this polytope