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

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

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

```

to this polytope