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

Atlas Canonical Name : {3,2,2,4}*96
if this polytope has a name.
Group : SmallGroup(96,209)
Rank : 5
Schlafli Type : {3,2,2,4}
Number of vertices, edges, etc : 3, 3, 2, 4, 4
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,2,2,4,2} of size 192
{3,2,2,4,3} of size 288
{3,2,2,4,4} of size 384
{3,2,2,4,6} of size 576
{3,2,2,4,3} of size 576
{3,2,2,4,6} of size 576
{3,2,2,4,6} of size 576
{3,2,2,4,8} of size 768
{3,2,2,4,8} of size 768
{3,2,2,4,4} of size 768
{3,2,2,4,9} of size 864
{3,2,2,4,4} of size 864
{3,2,2,4,6} of size 864
{3,2,2,4,10} of size 960
{3,2,2,4,12} of size 1152
{3,2,2,4,12} of size 1152
{3,2,2,4,12} of size 1152
{3,2,2,4,6} of size 1152
{3,2,2,4,14} of size 1344
{3,2,2,4,5} of size 1440
{3,2,2,4,6} of size 1440
{3,2,2,4,15} of size 1440
{3,2,2,4,18} of size 1728
{3,2,2,4,9} of size 1728
{3,2,2,4,18} of size 1728
{3,2,2,4,18} of size 1728
{3,2,2,4,4} of size 1728
{3,2,2,4,6} of size 1728
{3,2,2,4,20} of size 1920
{3,2,2,4,5} of size 1920
Vertex Figure Of :
{2,3,2,2,4} of size 192
{3,3,2,2,4} of size 384
{4,3,2,2,4} of size 384
{6,3,2,2,4} of size 576
{4,3,2,2,4} of size 768
{6,3,2,2,4} of size 768
{5,3,2,2,4} of size 960
{6,3,2,2,4} of size 1728
{5,3,2,2,4} of size 1920
{10,3,2,2,4} of size 1920
{10,3,2,2,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,2,4,4}*192, {3,2,2,8}*192, {6,2,2,4}*192
3-fold covers : {9,2,2,4}*288, {3,2,2,12}*288, {3,2,6,4}*288a, {3,6,2,4}*288
4-fold covers : {3,2,4,8}*384a, {3,2,8,4}*384a, {3,2,4,8}*384b, {3,2,8,4}*384b, {3,2,4,4}*384, {3,2,2,16}*384, {12,2,2,4}*384, {6,2,4,4}*384, {6,4,2,4}*384a, {6,2,2,8}*384, {3,4,2,4}*384
5-fold covers : {3,2,2,20}*480, {3,2,10,4}*480, {15,2,2,4}*480
6-fold covers : {9,2,4,4}*576, {9,2,2,8}*576, {18,2,2,4}*576, {3,2,4,12}*576a, {3,2,12,4}*576a, {3,2,2,24}*576, {3,2,6,8}*576, {3,6,2,8}*576, {3,6,4,4}*576, {6,2,2,12}*576, {6,2,6,4}*576a, {6,6,2,4}*576a, {6,6,2,4}*576c
7-fold covers : {3,2,2,28}*672, {3,2,14,4}*672, {21,2,2,4}*672
8-fold covers : {3,2,4,8}*768a, {3,2,8,4}*768a, {3,2,8,8}*768a, {3,2,8,8}*768b, {3,2,8,8}*768c, {3,2,8,8}*768d, {3,2,4,16}*768a, {3,2,16,4}*768a, {3,2,4,16}*768b, {3,2,16,4}*768b, {3,2,4,4}*768, {3,2,4,8}*768b, {3,2,8,4}*768b, {3,2,2,32}*768, {6,4,4,4}*768, {12,2,4,4}*768, {12,4,2,4}*768a, {6,2,4,8}*768a, {6,2,8,4}*768a, {6,2,4,8}*768b, {6,2,8,4}*768b, {6,2,4,4}*768, {6,4,2,8}*768a, {6,8,2,4}*768, {12,2,2,8}*768, {24,2,2,4}*768, {6,2,2,16}*768, {3,4,4,4}*768b, {3,4,2,8}*768, {3,8,2,4}*768, {6,4,2,4}*768
9-fold covers : {27,2,2,4}*864, {3,2,2,36}*864, {9,2,2,12}*864, {3,2,18,4}*864a, {9,2,6,4}*864a, {9,6,2,4}*864, {3,6,6,4}*864a, {3,6,2,4}*864, {3,2,6,12}*864a, {3,2,6,12}*864b, {3,6,2,12}*864, {3,2,6,12}*864c, {3,6,6,4}*864d, {3,2,6,4}*864
10-fold covers : {3,2,4,20}*960, {3,2,20,4}*960, {3,2,2,40}*960, {3,2,10,8}*960, {15,2,4,4}*960, {15,2,2,8}*960, {6,2,2,20}*960, {6,2,10,4}*960, {6,10,2,4}*960, {30,2,2,4}*960
11-fold covers : {3,2,2,44}*1056, {3,2,22,4}*1056, {33,2,2,4}*1056
12-fold covers : {9,2,4,8}*1152a, {9,2,8,4}*1152a, {3,6,4,8}*1152a, {3,2,8,12}*1152a, {3,2,12,8}*1152a, {3,6,8,4}*1152a, {3,2,4,24}*1152a, {3,2,24,4}*1152a, {9,2,4,8}*1152b, {9,2,8,4}*1152b, {3,6,4,8}*1152b, {3,2,8,12}*1152b, {3,2,12,8}*1152b, {3,6,8,4}*1152b, {3,2,4,24}*1152b, {3,2,24,4}*1152b, {9,2,4,4}*1152, {3,6,4,4}*1152, {3,2,4,12}*1152a, {3,2,12,4}*1152a, {9,2,2,16}*1152, {3,2,6,16}*1152, {3,6,2,16}*1152, {3,2,2,48}*1152, {18,2,4,4}*1152, {6,6,4,4}*1152b, {6,6,4,4}*1152c, {6,2,4,12}*1152a, {6,2,12,4}*1152a, {18,4,2,4}*1152a, {36,2,2,4}*1152, {6,4,6,4}*1152a, {6,12,2,4}*1152a, {6,4,2,12}*1152a, {6,12,2,4}*1152b, {12,2,6,4}*1152a, {12,6,2,4}*1152b, {12,6,2,4}*1152c, {12,2,2,12}*1152, {18,2,2,8}*1152, {6,2,6,8}*1152, {6,6,2,8}*1152a, {6,6,2,8}*1152c, {6,2,2,24}*1152, {9,4,2,4}*1152, {3,2,4,12}*1152b, {3,4,2,12}*1152, {3,4,6,4}*1152a, {3,2,6,4}*1152b, {3,2,6,12}*1152a, {3,6,2,4}*1152, {3,12,2,4}*1152
13-fold covers : {3,2,2,52}*1248, {3,2,26,4}*1248, {39,2,2,4}*1248
14-fold covers : {3,2,4,28}*1344, {3,2,28,4}*1344, {3,2,2,56}*1344, {3,2,14,8}*1344, {21,2,4,4}*1344, {21,2,2,8}*1344, {6,2,2,28}*1344, {6,2,14,4}*1344, {6,14,2,4}*1344, {42,2,2,4}*1344
15-fold covers : {9,2,2,20}*1440, {9,2,10,4}*1440, {45,2,2,4}*1440, {3,2,10,12}*1440, {3,2,6,20}*1440a, {3,6,2,20}*1440, {3,6,10,4}*1440, {15,2,2,12}*1440, {3,2,2,60}*1440, {3,2,30,4}*1440a, {15,2,6,4}*1440a, {15,6,2,4}*1440
17-fold covers : {3,2,2,68}*1632, {3,2,34,4}*1632, {51,2,2,4}*1632
18-fold covers : {27,2,4,4}*1728, {27,2,2,8}*1728, {54,2,2,4}*1728, {9,2,4,12}*1728a, {9,2,12,4}*1728a, {3,2,4,36}*1728a, {3,2,36,4}*1728a, {3,6,12,4}*1728a, {3,2,2,72}*1728, {9,2,2,24}*1728, {3,2,18,8}*1728, {9,2,6,8}*1728, {9,6,2,8}*1728, {3,6,6,8}*1728a, {3,6,2,8}*1728, {9,6,4,4}*1728, {3,6,4,4}*1728a, {18,2,2,12}*1728, {6,2,2,36}*1728, {6,2,18,4}*1728a, {6,18,2,4}*1728a, {18,2,6,4}*1728a, {18,6,2,4}*1728a, {18,6,2,4}*1728b, {6,6,6,4}*1728a, {6,6,2,4}*1728b, {6,6,2,4}*1728c, {3,2,6,24}*1728a, {3,2,6,24}*1728b, {3,6,2,24}*1728, {3,2,12,12}*1728a, {3,2,12,12}*1728b, {3,2,12,12}*1728c, {3,6,4,12}*1728, {3,2,6,24}*1728c, {3,6,6,8}*1728b, {3,6,12,4}*1728d, {3,2,6,8}*1728, {3,6,4,4}*1728b, {3,2,4,4}*1728, {3,2,4,12}*1728, {3,2,12,4}*1728, {6,2,6,12}*1728a, {6,2,6,12}*1728b, {6,6,2,12}*1728a, {6,6,2,12}*1728c, {6,6,6,4}*1728d, {6,6,6,4}*1728e, {6,6,2,4}*1728d, {6,6,6,4}*1728g, {6,2,6,12}*1728c, {6,6,6,4}*1728i, {6,2,6,4}*1728
19-fold covers : {3,2,2,76}*1824, {3,2,38,4}*1824, {57,2,2,4}*1824
20-fold covers : {15,2,4,8}*1920a, {15,2,8,4}*1920a, {3,2,8,20}*1920a, {3,2,20,8}*1920a, {3,2,4,40}*1920a, {3,2,40,4}*1920a, {15,2,4,8}*1920b, {15,2,8,4}*1920b, {3,2,8,20}*1920b, {3,2,20,8}*1920b, {3,2,4,40}*1920b, {3,2,40,4}*1920b, {15,2,4,4}*1920, {3,2,4,20}*1920, {3,2,20,4}*1920, {15,2,2,16}*1920, {3,2,10,16}*1920, {3,2,2,80}*1920, {30,2,4,4}*1920, {6,10,4,4}*1920, {6,2,4,20}*1920, {6,2,20,4}*1920, {30,4,2,4}*1920a, {60,2,2,4}*1920, {6,4,10,4}*1920, {12,2,10,4}*1920, {12,10,2,4}*1920, {6,4,2,20}*1920a, {6,20,2,4}*1920a, {12,2,2,20}*1920, {30,2,2,8}*1920, {6,2,10,8}*1920, {6,10,2,8}*1920, {6,2,2,40}*1920, {3,4,2,20}*1920, {3,4,10,4}*1920, {15,4,2,4}*1920
Permutation Representation (GAP) :
```s0 := (2,3);;
s1 := (1,2);;
s2 := (4,5);;
s3 := (7,8);;
s4 := (6,7)(8,9);;
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*s3*s4 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(9)!(2,3);
s1 := Sym(9)!(1,2);
s2 := Sym(9)!(4,5);
s3 := Sym(9)!(7,8);
s4 := Sym(9)!(6,7)(8,9);
poly := sub<Sym(9)|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*s3*s4 >;

```

to this polytope