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# Polytope of Type {2,3,4}

Atlas Canonical Name : {2,3,4}*48
if this polytope has a name.
Group : SmallGroup(48,48)
Rank : 4
Schlafli Type : {2,3,4}
Number of vertices, edges, etc : 2, 3, 6, 4
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Locally Projective
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,3,4,2} of size 96
{2,3,4,4} of size 384
{2,3,4,4} of size 768
{2,3,4,6} of size 1440
Vertex Figure Of :
{2,2,3,4} of size 96
{3,2,3,4} of size 144
{4,2,3,4} of size 192
{5,2,3,4} of size 240
{6,2,3,4} of size 288
{7,2,3,4} of size 336
{8,2,3,4} of size 384
{9,2,3,4} of size 432
{10,2,3,4} of size 480
{11,2,3,4} of size 528
{12,2,3,4} of size 576
{13,2,3,4} of size 624
{14,2,3,4} of size 672
{15,2,3,4} of size 720
{16,2,3,4} of size 768
{17,2,3,4} of size 816
{18,2,3,4} of size 864
{19,2,3,4} of size 912
{20,2,3,4} of size 960
{21,2,3,4} of size 1008
{22,2,3,4} of size 1056
{23,2,3,4} of size 1104
{24,2,3,4} of size 1152
{25,2,3,4} of size 1200
{26,2,3,4} of size 1248
{27,2,3,4} of size 1296
{28,2,3,4} of size 1344
{29,2,3,4} of size 1392
{30,2,3,4} of size 1440
{31,2,3,4} of size 1488
{33,2,3,4} of size 1584
{34,2,3,4} of size 1632
{35,2,3,4} of size 1680
{36,2,3,4} of size 1728
{37,2,3,4} of size 1776
{38,2,3,4} of size 1824
{39,2,3,4} of size 1872
{40,2,3,4} of size 1920
{41,2,3,4} of size 1968
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
3-fold covers : {2,9,4}*144, {6,3,4}*144
4-fold covers : {2,12,4}*192b, {2,12,4}*192c, {4,6,4}*192b, {2,3,8}*192, {2,6,4}*192, {4,3,4}*192b
5-fold covers : {2,15,4}*240
6-fold covers : {2,9,4}*288, {2,18,4}*288b, {2,18,4}*288c, {6,3,4}*288, {2,3,12}*288, {6,6,4}*288d, {2,6,12}*288d, {6,6,4}*288e, {6,6,4}*288f
7-fold covers : {2,21,4}*336
8-fold covers : {4,12,4}*384b, {4,12,4}*384c, {2,6,4}*384a, {2,3,8}*384, {2,6,8}*384a, {2,24,4}*384c, {2,24,4}*384d, {8,6,4}*384b, {2,12,4}*384b, {4,6,4}*384a, {2,6,4}*384b, {2,12,4}*384c, {2,6,8}*384b, {2,6,8}*384c, {8,3,4}*384, {4,3,4}*384, {4,6,4}*384d, {4,6,4}*384e
9-fold covers : {2,27,4}*432, {6,9,4}*432, {6,3,4}*432
10-fold covers : {10,6,4}*480b, {2,6,20}*480b, {2,15,4}*480, {2,30,4}*480b, {2,30,4}*480c
11-fold covers : {2,33,4}*528
12-fold covers : {2,36,4}*576b, {2,36,4}*576c, {4,18,4}*576b, {2,9,8}*576, {2,18,4}*576, {4,9,4}*576b, {6,12,4}*576d, {6,12,4}*576e, {6,12,4}*576f, {6,12,4}*576g, {12,6,4}*576d, {2,3,24}*576, {6,3,8}*576, {12,6,4}*576e, {6,3,4}*576, {6,6,4}*576a, {6,6,4}*576b, {2,6,12}*576a, {2,6,12}*576b, {12,3,4}*576
13-fold covers : {2,39,4}*624
14-fold covers : {2,6,28}*672b, {14,6,4}*672b, {2,21,4}*672, {2,42,4}*672b, {2,42,4}*672c
15-fold covers : {2,45,4}*720, {6,15,4}*720
16-fold covers : {2,3,8}*768, {2,6,8}*768a, {2,12,8}*768c, {2,12,8}*768d, {2,12,4}*768b, {2,12,4}*768c, {4,6,4}*768a, {4,24,4}*768e, {4,24,4}*768f, {4,12,4}*768c, {4,24,4}*768i, {4,24,4}*768j, {8,12,4}*768c, {8,12,4}*768d, {8,12,4}*768e, {2,6,8}*768b, {2,6,8}*768c, {2,48,4}*768c, {2,48,4}*768d, {16,6,4}*768b, {4,12,4}*768e, {2,12,4}*768d, {4,6,4}*768c, {4,12,4}*768g, {2,6,8}*768d, {2,6,8}*768e, {2,6,4}*768a, {2,12,8}*768e, {2,12,8}*768f, {2,24,4}*768c, {2,24,4}*768d, {2,6,8}*768f, {2,12,8}*768g, {2,12,8}*768h, {8,6,4}*768a, {2,6,8}*768g, {4,6,8}*768b, {4,6,8}*768c, {2,6,4}*768b, {2,24,4}*768e, {2,12,4}*768e, {2,24,4}*768f, {8,3,4}*768a, {4,6,4}*768h, {4,12,4}*768j, {4,12,4}*768k, {4,12,4}*768o, {4,12,4}*768p, {4,3,8}*768b, {8,3,4}*768b, {8,6,4}*768e, {8,6,4}*768f, {8,6,4}*768g, {4,3,8}*768c, {8,3,4}*768c, {8,6,4}*768h, {4,3,4}*768, {4,6,4}*768j, {4,6,4}*768k, {4,6,4}*768l
17-fold covers : {2,51,4}*816
18-fold covers : {2,27,4}*864, {2,54,4}*864b, {2,54,4}*864c, {2,6,36}*864c, {18,6,4}*864c, {6,9,4}*864, {6,18,4}*864c, {2,9,12}*864, {6,18,4}*864d, {6,18,4}*864e, {2,18,12}*864c, {2,3,12}*864, {2,6,12}*864d, {6,3,4}*864, {6,6,4}*864e, {6,6,4}*864f, {6,6,4}*864g, {6,3,12}*864, {6,6,4}*864i, {6,6,12}*864h
19-fold covers : {2,57,4}*912
20-fold covers : {10,12,4}*960b, {10,12,4}*960c, {20,6,4}*960b, {2,60,4}*960b, {2,60,4}*960c, {4,30,4}*960b, {2,15,8}*960, {10,6,4}*960e, {2,6,20}*960c, {2,30,4}*960, {4,15,4}*960b
21-fold covers : {2,63,4}*1008, {6,21,4}*1008
22-fold covers : {2,6,44}*1056b, {22,6,4}*1056b, {2,33,4}*1056, {2,66,4}*1056b, {2,66,4}*1056c
23-fold covers : {2,69,4}*1104
24-fold covers : {4,36,4}*1152b, {4,36,4}*1152c, {2,18,4}*1152a, {2,9,8}*1152, {2,18,8}*1152a, {2,72,4}*1152c, {2,72,4}*1152d, {8,18,4}*1152b, {2,36,4}*1152b, {4,18,4}*1152a, {2,18,4}*1152b, {2,36,4}*1152c, {2,18,8}*1152b, {2,18,8}*1152c, {8,9,4}*1152, {2,3,24}*1152, {2,6,24}*1152a, {6,6,4}*1152a, {6,6,4}*1152b, {6,3,8}*1152, {6,6,8}*1152a, {6,24,4}*1152g, {6,24,4}*1152h, {6,24,4}*1152i, {6,24,4}*1152j, {24,6,4}*1152d, {12,12,4}*1152d, {12,12,4}*1152e, {12,12,4}*1152f, {12,12,4}*1152g, {24,6,4}*1152e, {4,9,4}*1152, {4,18,4}*1152d, {4,18,4}*1152e, {12,3,4}*1152a, {6,12,4}*1152e, {6,12,4}*1152f, {2,12,12}*1152d, {2,12,12}*1152e, {12,6,4}*1152a, {2,6,12}*1152b, {2,12,12}*1152h, {4,6,12}*1152b, {4,6,12}*1152c, {6,6,4}*1152c, {6,6,4}*1152d, {6,12,4}*1152g, {6,12,4}*1152h, {2,6,24}*1152b, {2,6,24}*1152c, {2,6,24}*1152d, {6,6,8}*1152b, {6,6,8}*1152c, {2,6,24}*1152e, {6,6,8}*1152d, {6,6,8}*1152e, {2,6,12}*1152f, {12,6,4}*1152d, {2,12,12}*1152j, {24,3,4}*1152, {2,3,12}*1152, {6,3,4}*1152b, {6,6,4}*1152g, {2,12,12}*1152l, {6,6,4}*1152h, {6,12,4}*1152l, {4,3,12}*1152b, {4,6,12}*1152g, {12,3,4}*1152b, {12,6,4}*1152f, {12,6,4}*1152g, {12,6,4}*1152h
25-fold covers : {2,75,4}*1200, {10,15,4}*1200, {10,3,4}*1200
26-fold covers : {2,6,52}*1248b, {26,6,4}*1248b, {2,39,4}*1248, {2,78,4}*1248b, {2,78,4}*1248c
27-fold covers : {2,81,4}*1296, {6,27,4}*1296, {18,9,4}*1296, {18,3,4}*1296, {6,3,4}*1296a, {6,9,4}*1296a, {6,9,4}*1296b, {6,9,4}*1296c, {6,9,4}*1296d, {2,3,12}*1296, {2,9,4}*1296, {2,9,12}*1296a, {2,9,12}*1296b
28-fold covers : {14,12,4}*1344b, {14,12,4}*1344c, {28,6,4}*1344b, {2,84,4}*1344b, {2,84,4}*1344c, {4,42,4}*1344b, {2,21,8}*1344, {14,6,4}*1344, {2,6,28}*1344, {2,42,4}*1344, {4,21,4}*1344b
29-fold covers : {2,87,4}*1392
30-fold covers : {10,18,4}*1440b, {2,18,20}*1440b, {2,45,4}*1440, {2,90,4}*1440b, {2,90,4}*1440c, {6,6,20}*1440d, {30,6,4}*1440d, {6,15,4}*1440b, {6,30,4}*1440d, {6,30,4}*1440e, {6,30,4}*1440f, {2,15,12}*1440, {2,30,12}*1440d, {2,6,60}*1440d, {30,6,4}*1440e
31-fold covers : {2,93,4}*1488
33-fold covers : {2,99,4}*1584, {6,33,4}*1584
34-fold covers : {2,6,68}*1632b, {34,6,4}*1632b, {2,51,4}*1632, {2,102,4}*1632b, {2,102,4}*1632c
35-fold covers : {2,105,4}*1680
36-fold covers : {2,108,4}*1728b, {2,108,4}*1728c, {4,54,4}*1728b, {2,27,8}*1728, {2,54,4}*1728, {4,27,4}*1728b, {18,12,4}*1728c, {18,12,4}*1728d, {36,6,4}*1728c, {6,36,4}*1728c, {6,36,4}*1728d, {6,36,4}*1728e, {6,36,4}*1728f, {12,18,4}*1728c, {6,12,4}*1728d, {6,12,4}*1728e, {6,12,4}*1728f, {6,12,4}*1728g, {12,6,4}*1728d, {2,9,24}*1728, {2,3,24}*1728, {6,9,8}*1728, {6,3,8}*1728, {12,18,4}*1728d, {12,6,4}*1728e, {6,9,4}*1728, {6,3,4}*1728, {18,6,4}*1728, {2,6,36}*1728, {6,18,4}*1728a, {6,18,4}*1728b, {2,18,12}*1728a, {2,18,12}*1728b, {6,6,4}*1728a, {6,6,4}*1728b, {2,6,12}*1728a, {2,6,12}*1728b, {12,9,4}*1728, {12,3,4}*1728, {6,3,24}*1728, {6,12,4}*1728l, {6,12,4}*1728m, {12,6,4}*1728j, {4,6,4}*1728e, {4,12,4}*1728e, {6,12,4}*1728s, {2,12,12}*1728n, {6,6,4}*1728c, {6,6,12}*1728a, {6,6,12}*1728b, {6,6,12}*1728c, {6,6,12}*1728d, {2,6,12}*1728c
37-fold covers : {2,111,4}*1776
38-fold covers : {2,6,76}*1824b, {38,6,4}*1824b, {2,57,4}*1824, {2,114,4}*1824b, {2,114,4}*1824c
39-fold covers : {2,117,4}*1872, {6,39,4}*1872
40-fold covers : {2,6,40}*1920a, {10,6,4}*1920a, {10,24,4}*1920c, {10,24,4}*1920d, {40,6,4}*1920b, {20,12,4}*1920b, {20,12,4}*1920c, {4,60,4}*1920b, {4,60,4}*1920c, {2,30,4}*1920a, {2,15,8}*1920a, {2,30,8}*1920a, {2,120,4}*1920c, {2,120,4}*1920d, {8,30,4}*1920b, {10,12,4}*1920b, {2,12,20}*1920b, {20,6,4}*1920a, {2,6,20}*1920a, {4,6,20}*1920b, {10,6,4}*1920b, {10,12,4}*1920c, {2,6,40}*1920b, {10,6,8}*1920a, {2,6,40}*1920c, {10,6,8}*1920b, {2,12,20}*1920c, {2,60,4}*1920b, {4,30,4}*1920a, {2,30,4}*1920b, {2,60,4}*1920c, {2,30,8}*1920b, {2,30,8}*1920c, {8,15,4}*1920, {4,6,20}*1920d, {20,6,4}*1920d, {4,15,4}*1920c, {4,30,4}*1920d, {4,30,4}*1920e
41-fold covers : {2,123,4}*1968
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (5,6);;
s2 := (4,5);;
s3 := (3,4)(5,6);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s3*s2*s1*s3*s2*s1*s3*s2 ];;
poly := F / rels;;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s3*s2*s1*s3*s2*s1*s3*s2 >;

```

to this polytope