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# Polytope of Type {4,2,3}

Atlas Canonical Name : {4,2,3}*48
if this polytope has a name.
Group : SmallGroup(48,38)
Rank : 4
Schlafli Type : {4,2,3}
Number of vertices, edges, etc : 4, 4, 3, 3
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Projective
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,2,3,2} of size 96
{4,2,3,3} of size 192
{4,2,3,4} of size 192
{4,2,3,6} of size 288
{4,2,3,4} of size 384
{4,2,3,6} of size 384
{4,2,3,5} of size 480
{4,2,3,8} of size 768
{4,2,3,12} of size 768
{4,2,3,6} of size 864
{4,2,3,5} of size 960
{4,2,3,10} of size 960
{4,2,3,10} of size 960
{4,2,3,6} of size 1152
{4,2,3,12} of size 1152
{4,2,3,10} of size 1920
Vertex Figure Of :
{2,4,2,3} of size 96
{3,4,2,3} of size 144
{4,4,2,3} of size 192
{6,4,2,3} of size 288
{3,4,2,3} of size 288
{6,4,2,3} of size 288
{6,4,2,3} of size 288
{8,4,2,3} of size 384
{8,4,2,3} of size 384
{4,4,2,3} of size 384
{9,4,2,3} of size 432
{4,4,2,3} of size 432
{6,4,2,3} of size 432
{10,4,2,3} of size 480
{12,4,2,3} of size 576
{12,4,2,3} of size 576
{12,4,2,3} of size 576
{6,4,2,3} of size 576
{14,4,2,3} of size 672
{5,4,2,3} of size 720
{6,4,2,3} of size 720
{15,4,2,3} of size 720
{8,4,2,3} of size 768
{16,4,2,3} of size 768
{16,4,2,3} of size 768
{4,4,2,3} of size 768
{8,4,2,3} of size 768
{18,4,2,3} of size 864
{9,4,2,3} of size 864
{18,4,2,3} of size 864
{18,4,2,3} of size 864
{4,4,2,3} of size 864
{6,4,2,3} of size 864
{20,4,2,3} of size 960
{5,4,2,3} of size 960
{21,4,2,3} of size 1008
{22,4,2,3} of size 1056
{24,4,2,3} of size 1152
{24,4,2,3} of size 1152
{12,4,2,3} of size 1152
{6,4,2,3} of size 1152
{24,4,2,3} of size 1152
{24,4,2,3} of size 1152
{12,4,2,3} of size 1152
{6,4,2,3} of size 1152
{12,4,2,3} of size 1152
{4,4,2,3} of size 1200
{10,4,2,3} of size 1200
{26,4,2,3} of size 1248
{27,4,2,3} of size 1296
{6,4,2,3} of size 1296
{12,4,2,3} of size 1296
{28,4,2,3} of size 1344
{30,4,2,3} of size 1440
{5,4,2,3} of size 1440
{6,4,2,3} of size 1440
{6,4,2,3} of size 1440
{6,4,2,3} of size 1440
{10,4,2,3} of size 1440
{10,4,2,3} of size 1440
{15,4,2,3} of size 1440
{30,4,2,3} of size 1440
{30,4,2,3} of size 1440
{33,4,2,3} of size 1584
{34,4,2,3} of size 1632
{36,4,2,3} of size 1728
{36,4,2,3} of size 1728
{36,4,2,3} of size 1728
{18,4,2,3} of size 1728
{4,4,2,3} of size 1728
{12,4,2,3} of size 1728
{38,4,2,3} of size 1824
{39,4,2,3} of size 1872
{40,4,2,3} of size 1920
{40,4,2,3} of size 1920
{20,4,2,3} of size 1920
{5,4,2,3} of size 1920
{10,4,2,3} of size 1920
{10,4,2,3} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,2,3}*96, {4,2,6}*96
3-fold covers : {4,2,9}*144, {12,2,3}*144, {4,6,3}*144
4-fold covers : {16,2,3}*192, {4,2,12}*192, {4,4,6}*192, {8,2,6}*192, {4,4,3}*192b
5-fold covers : {20,2,3}*240, {4,2,15}*240
6-fold covers : {8,2,9}*288, {4,2,18}*288, {24,2,3}*288, {8,6,3}*288, {12,2,6}*288, {4,6,6}*288a, {4,6,6}*288c
7-fold covers : {28,2,3}*336, {4,2,21}*336
8-fold covers : {32,2,3}*384, {4,4,12}*384, {4,2,24}*384, {8,2,12}*384, {4,8,6}*384a, {8,4,6}*384a, {4,8,6}*384b, {8,4,6}*384b, {4,4,6}*384a, {16,2,6}*384, {4,8,3}*384, {8,4,3}*384, {4,4,6}*384d
9-fold covers : {4,2,27}*432, {36,2,3}*432, {12,2,9}*432, {12,6,3}*432a, {4,6,9}*432, {4,6,3}*432a, {12,6,3}*432b, {4,6,3}*432b
10-fold covers : {40,2,3}*480, {8,2,15}*480, {20,2,6}*480, {4,10,6}*480, {4,2,30}*480
11-fold covers : {44,2,3}*528, {4,2,33}*528
12-fold covers : {16,2,9}*576, {4,2,36}*576, {4,4,18}*576, {8,2,18}*576, {48,2,3}*576, {16,6,3}*576, {4,4,9}*576b, {12,2,12}*576, {4,12,6}*576a, {12,4,6}*576, {4,6,12}*576a, {24,2,6}*576, {8,6,6}*576a, {4,6,12}*576b, {8,6,6}*576c, {4,12,6}*576c, {12,4,3}*576, {4,6,3}*576a, {4,12,3}*576
13-fold covers : {52,2,3}*624, {4,2,39}*624
14-fold covers : {56,2,3}*672, {8,2,21}*672, {28,2,6}*672, {4,14,6}*672, {4,2,42}*672
15-fold covers : {20,2,9}*720, {4,2,45}*720, {20,6,3}*720, {12,2,15}*720, {60,2,3}*720, {4,6,15}*720
16-fold covers : {64,2,3}*768, {4,8,6}*768a, {8,4,6}*768a, {8,8,6}*768a, {8,8,6}*768b, {8,8,6}*768c, {8,8,6}*768d, {8,2,24}*768, {8,4,12}*768a, {4,4,24}*768a, {8,4,12}*768b, {4,4,24}*768b, {4,8,12}*768a, {4,4,12}*768a, {4,4,12}*768b, {4,8,12}*768b, {4,8,12}*768c, {4,8,12}*768d, {4,16,6}*768a, {16,4,6}*768a, {4,16,6}*768b, {16,4,6}*768b, {4,4,6}*768a, {4,8,6}*768b, {8,4,6}*768b, {16,2,12}*768, {4,2,48}*768, {32,2,6}*768, {8,8,3}*768, {4,4,3}*768a, {4,8,3}*768c, {4,8,3}*768d, {16,4,3}*768, {4,4,6}*768e, {4,4,12}*768e, {4,4,12}*768f, {4,8,6}*768c, {8,4,6}*768c, {4,8,6}*768d
17-fold covers : {68,2,3}*816, {4,2,51}*816
18-fold covers : {8,2,27}*864, {4,2,54}*864, {72,2,3}*864, {24,2,9}*864, {24,6,3}*864a, {8,6,9}*864, {8,6,3}*864a, {36,2,6}*864, {12,2,18}*864, {12,6,6}*864a, {4,6,18}*864a, {4,18,6}*864a, {4,6,6}*864b, {4,6,18}*864b, {4,6,6}*864c, {24,6,3}*864b, {8,6,3}*864b, {12,6,6}*864b, {12,6,6}*864d, {12,6,6}*864e, {4,6,6}*864h, {12,6,6}*864f, {4,6,6}*864j, {4,6,6}*864k
19-fold covers : {76,2,3}*912, {4,2,57}*912
20-fold covers : {80,2,3}*960, {16,2,15}*960, {20,2,12}*960, {4,20,6}*960, {20,4,6}*960, {4,10,12}*960, {40,2,6}*960, {8,10,6}*960, {4,2,60}*960, {4,4,30}*960, {8,2,30}*960, {20,4,3}*960, {4,4,15}*960b
21-fold covers : {28,2,9}*1008, {4,2,63}*1008, {28,6,3}*1008, {12,2,21}*1008, {84,2,3}*1008, {4,6,21}*1008
22-fold covers : {88,2,3}*1056, {8,2,33}*1056, {44,2,6}*1056, {4,22,6}*1056, {4,2,66}*1056
23-fold covers : {92,2,3}*1104, {4,2,69}*1104
24-fold covers : {32,2,9}*1152, {32,6,3}*1152, {96,2,3}*1152, {4,4,36}*1152, {4,12,12}*1152b, {4,12,12}*1152c, {12,4,12}*1152, {4,8,18}*1152a, {8,4,18}*1152a, {8,12,6}*1152b, {12,8,6}*1152a, {4,24,6}*1152a, {8,12,6}*1152c, {4,24,6}*1152c, {24,4,6}*1152a, {4,8,18}*1152b, {8,4,18}*1152b, {8,12,6}*1152e, {12,8,6}*1152b, {4,24,6}*1152d, {8,12,6}*1152f, {4,24,6}*1152f, {24,4,6}*1152b, {4,4,18}*1152a, {4,12,6}*1152b, {12,4,6}*1152a, {4,12,6}*1152c, {8,2,36}*1152, {4,2,72}*1152, {8,6,12}*1152b, {8,6,12}*1152c, {4,6,24}*1152b, {4,6,24}*1152c, {12,2,24}*1152, {24,2,12}*1152, {16,2,18}*1152, {16,6,6}*1152a, {16,6,6}*1152c, {48,2,6}*1152, {4,8,9}*1152, {8,4,9}*1152, {4,4,18}*1152d, {12,8,3}*1152, {24,4,3}*1152, {8,6,3}*1152, {8,12,3}*1152, {4,12,3}*1152b, {4,24,3}*1152, {12,4,6}*1152b, {12,4,6}*1152c, {4,6,6}*1152d, {4,6,12}*1152b, {4,12,6}*1152g, {12,6,6}*1152b, {4,6,6}*1152f, {4,12,6}*1152j
25-fold covers : {100,2,3}*1200, {4,2,75}*1200, {4,10,3}*1200, {20,2,15}*1200, {4,10,15}*1200
26-fold covers : {104,2,3}*1248, {8,2,39}*1248, {52,2,6}*1248, {4,26,6}*1248, {4,2,78}*1248
27-fold covers : {4,2,81}*1296, {36,2,9}*1296, {12,6,9}*1296a, {36,6,3}*1296a, {12,2,27}*1296, {108,2,3}*1296, {12,6,3}*1296a, {12,6,3}*1296b, {4,18,9}*1296, {4,6,9}*1296a, {4,6,27}*1296, {4,6,9}*1296b, {4,6,9}*1296c, {4,6,9}*1296d, {4,6,3}*1296a, {4,18,3}*1296, {36,6,3}*1296b, {12,6,9}*1296b, {12,6,3}*1296c, {12,6,3}*1296d, {12,6,3}*1296e, {4,6,9}*1296e, {4,6,3}*1296b, {12,6,3}*1296f
28-fold covers : {112,2,3}*1344, {16,2,21}*1344, {28,2,12}*1344, {4,14,12}*1344, {4,28,6}*1344, {28,4,6}*1344, {56,2,6}*1344, {8,14,6}*1344, {4,2,84}*1344, {4,4,42}*1344, {8,2,42}*1344, {28,4,3}*1344, {4,4,21}*1344b
29-fold covers : {116,2,3}*1392, {4,2,87}*1392
30-fold covers : {40,2,9}*1440, {8,2,45}*1440, {20,2,18}*1440, {4,10,18}*1440, {4,2,90}*1440, {40,6,3}*1440, {24,2,15}*1440, {120,2,3}*1440, {8,6,15}*1440, {12,10,6}*1440, {20,6,6}*1440a, {20,6,6}*1440c, {4,30,6}*1440a, {12,2,30}*1440, {60,2,6}*1440, {4,6,30}*1440b, {4,30,6}*1440b, {4,6,30}*1440c
31-fold covers : {124,2,3}*1488, {4,2,93}*1488
33-fold covers : {44,2,9}*1584, {4,2,99}*1584, {44,6,3}*1584, {12,2,33}*1584, {132,2,3}*1584, {4,6,33}*1584
34-fold covers : {136,2,3}*1632, {8,2,51}*1632, {68,2,6}*1632, {4,34,6}*1632, {4,2,102}*1632
35-fold covers : {28,2,15}*1680, {20,2,21}*1680, {140,2,3}*1680, {4,2,105}*1680
36-fold covers : {16,2,27}*1728, {4,2,108}*1728, {4,4,54}*1728, {8,2,54}*1728, {144,2,3}*1728, {48,2,9}*1728, {48,6,3}*1728a, {16,6,9}*1728, {16,6,3}*1728a, {4,4,27}*1728b, {12,2,36}*1728, {36,2,12}*1728, {12,6,12}*1728a, {4,6,36}*1728a, {4,18,12}*1728a, {4,12,18}*1728a, {12,4,18}*1728, {4,36,6}*1728a, {36,4,6}*1728, {4,6,12}*1728a, {4,12,6}*1728b, {12,12,6}*1728a, {72,2,6}*1728, {24,2,18}*1728, {24,6,6}*1728a, {8,6,18}*1728a, {8,18,6}*1728a, {8,6,6}*1728b, {4,6,36}*1728b, {4,6,12}*1728b, {8,6,18}*1728b, {8,6,6}*1728c, {4,12,18}*1728b, {4,12,6}*1728c, {48,6,3}*1728b, {36,4,3}*1728, {4,6,9}*1728a, {12,4,9}*1728, {12,12,3}*1728a, {4,12,9}*1728, {4,6,3}*1728a, {4,12,3}*1728a, {16,6,3}*1728b, {24,6,6}*1728b, {24,6,6}*1728d, {24,6,6}*1728e, {12,6,12}*1728b, {12,6,12}*1728c, {12,6,12}*1728e, {12,6,12}*1728f, {12,12,6}*1728b, {12,12,6}*1728c, {12,12,6}*1728f, {8,6,6}*1728e, {24,6,6}*1728f, {4,12,6}*1728j, {12,12,6}*1728g, {4,6,12}*1728h, {12,6,3}*1728, {12,12,3}*1728b, {8,6,6}*1728f, {4,6,12}*1728k, {4,6,12}*1728l, {8,6,6}*1728g, {4,4,6}*1728b, {4,4,6}*1728c, {4,12,6}*1728n, {4,12,6}*1728o, {12,4,6}*1728b, {4,4,12}*1728c, {4,6,12}*1728n, {4,12,3}*1728b
37-fold covers : {148,2,3}*1776, {4,2,111}*1776
38-fold covers : {152,2,3}*1824, {8,2,57}*1824, {76,2,6}*1824, {4,38,6}*1824, {4,2,114}*1824
39-fold covers : {52,2,9}*1872, {4,2,117}*1872, {52,6,3}*1872, {12,2,39}*1872, {156,2,3}*1872, {4,6,39}*1872
40-fold covers : {32,2,15}*1920, {160,2,3}*1920, {4,4,60}*1920, {4,20,12}*1920, {20,4,12}*1920, {4,8,30}*1920a, {8,4,30}*1920a, {8,20,6}*1920a, {20,8,6}*1920a, {4,40,6}*1920a, {40,4,6}*1920a, {4,8,30}*1920b, {8,4,30}*1920b, {8,20,6}*1920b, {20,8,6}*1920b, {4,40,6}*1920b, {40,4,6}*1920b, {4,4,30}*1920a, {4,20,6}*1920a, {20,4,6}*1920a, {8,2,60}*1920, {4,2,120}*1920, {8,10,12}*1920, {4,10,24}*1920, {40,2,12}*1920, {20,2,24}*1920, {16,2,30}*1920, {16,10,6}*1920, {80,2,6}*1920, {20,8,3}*1920, {40,4,3}*1920, {4,8,15}*1920, {8,4,15}*1920, {20,4,6}*1920b, {4,20,6}*1920c, {4,4,30}*1920d
41-fold covers : {164,2,3}*1968, {4,2,123}*1968
Permutation Representation (GAP) :
```s0 := (2,3);;
s1 := (1,2)(3,4);;
s2 := (6,7);;
s3 := (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*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```

to this polytope