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# Polytope of Type {4,3}

Atlas Canonical Name : {4,3}*24
Also Known As : hemicube, {4,3}3if this polytope has another name.
Group : SmallGroup(24,12)
Rank : 3
Schlafli Type : {4,3}
Number of vertices, edges, etc : 4, 6, 3
Order of s0s1s2 : 3
Order of s0s1s2s1 : 4
Special Properties :
Projective
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
{4,3,2} of size 48
{4,3,4} of size 96
{4,3,6} of size 144
{4,3,4} of size 192
{4,3,8} of size 384
{4,3,6} of size 432
{4,3,6} of size 576
{4,3,12} of size 576
{4,3,8} of size 768
{4,3,12} of size 1152
{4,3,24} of size 1152
{4,3,10} of size 1200
{4,3,18} of size 1296
{4,3,6} of size 1296
{4,3,10} of size 1440
{4,3,6} of size 1728
{4,3,12} of size 1728
Vertex Figure Of :
{2,4,3} of size 48
{4,4,3} of size 192
{4,4,3} of size 384
{6,4,3} of size 720
{8,4,3} of size 768
{8,4,3} of size 768
{6,4,3} of size 1296
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,3}*48, {4,6}*48b, {4,6}*48c
3-fold covers : {4,9}*72
4-fold covers : {4,12}*96b, {4,12}*96c, {8,3}*96, {4,6}*96
5-fold covers : {4,15}*120
6-fold covers : {4,9}*144, {4,18}*144b, {4,18}*144c, {12,3}*144, {12,6}*144d
7-fold covers : {4,21}*168
8-fold covers : {4,6}*192a, {8,3}*192, {8,6}*192a, {4,24}*192c, {4,24}*192d, {4,12}*192b, {4,6}*192b, {4,12}*192c, {8,6}*192b, {8,6}*192c
9-fold covers : {4,27}*216
10-fold covers : {20,6}*240b, {4,15}*240, {4,30}*240b, {4,30}*240c
11-fold covers : {4,33}*264
12-fold covers : {4,36}*288b, {4,36}*288c, {8,9}*288, {4,18}*288, {24,3}*288, {12,6}*288a, {12,6}*288b
13-fold covers : {4,39}*312
14-fold covers : {28,6}*336b, {4,21}*336, {4,42}*336b, {4,42}*336c
15-fold covers : {4,45}*360
16-fold covers : {4,12}*384b, {4,12}*384c, {8,3}*384, {8,6}*384a, {8,12}*384c, {8,12}*384d, {8,6}*384b, {8,6}*384c, {4,48}*384c, {4,48}*384d, {4,12}*384d, {8,12}*384e, {8,12}*384f, {4,6}*384a, {8,6}*384d, {8,6}*384e, {8,6}*384f, {8,12}*384g, {8,12}*384h, {4,24}*384c, {4,24}*384d, {8,6}*384g, {4,12}*384e, {4,24}*384e, {4,6}*384b, {4,24}*384f
17-fold covers : {4,51}*408
18-fold covers : {4,27}*432, {4,54}*432b, {4,54}*432c, {36,6}*432c, {12,9}*432, {12,18}*432c, {12,3}*432, {12,6}*432d
19-fold covers : {4,57}*456
20-fold covers : {4,60}*480b, {4,60}*480c, {8,15}*480, {20,6}*480c, {4,30}*480
21-fold covers : {4,63}*504
22-fold covers : {44,6}*528b, {4,33}*528, {4,66}*528b, {4,66}*528c
23-fold covers : {4,69}*552
24-fold covers : {4,18}*576a, {8,9}*576, {8,18}*576a, {4,72}*576c, {4,72}*576d, {4,36}*576b, {4,18}*576b, {4,36}*576c, {8,18}*576b, {8,18}*576c, {24,3}*576, {24,6}*576a, {12,12}*576f, {12,12}*576g, {12,6}*576b, {12,12}*576i, {24,6}*576b, {24,6}*576c, {24,6}*576d, {24,6}*576e, {12,6}*576f, {12,12}*576k, {12,3}*576, {12,12}*576l
25-fold covers : {4,75}*600
26-fold covers : {52,6}*624b, {4,39}*624, {4,78}*624b, {4,78}*624c
27-fold covers : {4,81}*648, {4,9}*648, {12,3}*648, {12,9}*648a, {12,9}*648b
28-fold covers : {4,84}*672b, {4,84}*672c, {8,21}*672, {28,6}*672, {4,42}*672
29-fold covers : {4,87}*696
30-fold covers : {20,18}*720b, {4,45}*720, {4,90}*720b, {4,90}*720c, {12,15}*720, {12,30}*720d, {60,6}*720d
31-fold covers : {4,93}*744
32-fold covers : {4,24}*768e, {4,24}*768f, {16,3}*768a, {16,3}*768b, {16,6}*768a, {8,12}*768e, {8,12}*768f, {8,12}*768g, {8,12}*768h, {4,24}*768g, {4,24}*768h, {8,6}*768a, {8,6}*768b, {8,6}*768c, {8,12}*768i, {8,12}*768j, {4,96}*768c, {4,96}*768d, {8,6}*768d, {8,12}*768k, {8,6}*768e, {8,6}*768f, {8,12}*768l, {8,6}*768g, {8,6}*768h, {8,6}*768i, {8,12}*768m, {8,12}*768n, {8,24}*768i, {8,24}*768j, {8,24}*768k, {8,24}*768l, {8,6}*768j, {8,24}*768m, {8,12}*768o, {8,24}*768n, {8,12}*768p, {8,24}*768o, {8,24}*768p, {4,12}*768b, {4,6}*768a, {4,12}*768c, {8,12}*768q, {8,12}*768r, {8,12}*768s, {4,24}*768i, {4,12}*768d, {8,12}*768t, {4,24}*768j, {8,12}*768u, {4,12}*768e, {4,24}*768k, {8,6}*768k, {8,12}*768v, {8,12}*768w, {4,12}*768f, {4,24}*768l, {8,6}*768l, {8,12}*768x, {8,6}*768m, {8,6}*768n, {4,6}*768b, {4,6}*768c, {4,12}*768g, {4,12}*768h, {4,48}*768c, {4,48}*768d, {16,6}*768b, {16,6}*768c
33-fold covers : {4,99}*792
34-fold covers : {68,6}*816b, {4,51}*816, {4,102}*816b, {4,102}*816c
35-fold covers : {4,105}*840
36-fold covers : {4,108}*864b, {4,108}*864c, {8,27}*864, {4,54}*864, {24,9}*864, {24,3}*864, {36,6}*864, {12,18}*864a, {12,18}*864b, {12,6}*864a, {12,6}*864b, {12,12}*864o, {12,6}*864c
37-fold covers : {4,111}*888
38-fold covers : {76,6}*912b, {4,57}*912, {4,114}*912b, {4,114}*912c
39-fold covers : {4,117}*936
40-fold covers : {40,6}*960c, {4,30}*960a, {8,15}*960a, {8,30}*960a, {4,120}*960c, {4,120}*960d, {20,12}*960b, {20,6}*960e, {40,6}*960d, {40,6}*960e, {20,12}*960c, {4,60}*960b, {4,30}*960b, {4,60}*960c, {8,30}*960b, {8,30}*960c
41-fold covers : {4,123}*984
42-fold covers : {28,18}*1008b, {4,63}*1008, {4,126}*1008b, {4,126}*1008c, {12,21}*1008, {12,42}*1008d, {84,6}*1008d
43-fold covers : {4,129}*1032
44-fold covers : {4,132}*1056b, {4,132}*1056c, {8,33}*1056, {44,6}*1056, {4,66}*1056
45-fold covers : {4,135}*1080
46-fold covers : {92,6}*1104b, {4,69}*1104, {4,138}*1104b, {4,138}*1104c
47-fold covers : {4,141}*1128
48-fold covers : {4,36}*1152b, {4,36}*1152c, {8,9}*1152, {8,18}*1152a, {8,36}*1152c, {8,36}*1152d, {8,18}*1152b, {8,18}*1152c, {4,144}*1152c, {4,144}*1152d, {4,36}*1152d, {8,36}*1152e, {8,36}*1152f, {4,18}*1152a, {8,18}*1152d, {8,18}*1152e, {8,18}*1152f, {8,36}*1152g, {8,36}*1152h, {4,72}*1152c, {4,72}*1152d, {8,18}*1152g, {4,36}*1152e, {4,72}*1152e, {4,18}*1152b, {4,72}*1152f, {24,3}*1152a, {24,6}*1152a, {24,12}*1152g, {24,12}*1152h, {24,6}*1152b, {24,6}*1152c, {24,12}*1152i, {24,12}*1152j, {24,12}*1152k, {24,12}*1152l, {24,12}*1152m, {24,6}*1152d, {24,12}*1152n, {12,6}*1152b, {12,6}*1152c, {24,6}*1152e, {24,6}*1152f, {12,24}*1152o, {12,24}*1152p, {12,24}*1152q, {12,24}*1152r, {24,6}*1152h, {12,6}*1152d, {12,24}*1152s, {12,12}*1152i, {12,24}*1152t, {12,12}*1152n, {12,12}*1152o, {24,6}*1152k, {24,6}*1152l, {24,12}*1152u, {24,12}*1152v, {12,12}*1152r, {12,24}*1152w, {12,6}*1152f, {12,24}*1152x, {12,3}*1152b, {12,6}*1152g, {12,24}*1152y, {12,24}*1152z, {24,3}*1152b, {24,12}*1152y, {24,12}*1152z, {24,3}*1152c, {12,6}*1152j, {12,12}*1152t
49-fold covers : {4,147}*1176
50-fold covers : {100,6}*1200b, {4,75}*1200, {4,150}*1200b, {4,150}*1200c, {20,15}*1200, {20,30}*1200d, {20,3}*1200, {20,6}*1200d
51-fold covers : {4,153}*1224
52-fold covers : {4,156}*1248b, {4,156}*1248c, {8,39}*1248, {52,6}*1248, {4,78}*1248
53-fold covers : {4,159}*1272
54-fold covers : {4,81}*1296, {4,162}*1296b, {4,162}*1296c, {108,6}*1296c, {12,27}*1296, {12,54}*1296c, {36,9}*1296, {36,18}*1296d, {36,6}*1296i, {36,3}*1296, {36,6}*1296j, {36,6}*1296k, {12,3}*1296a, {12,18}*1296i, {12,9}*1296a, {12,18}*1296j, {12,6}*1296e, {12,9}*1296b, {12,9}*1296c, {12,18}*1296k, {12,6}*1296f, {12,9}*1296d, {4,6}*1296b, {4,6}*1296c, {4,9}*1296a, {12,6}*1296p, {4,9}*1296b, {4,18}*1296c, {4,18}*1296d, {12,3}*1296b, {12,6}*1296q, {12,6}*1296r, {12,9}*1296e, {12,9}*1296f, {12,18}*1296m, {12,18}*1296n, {12,18}*1296o, {12,18}*1296p
55-fold covers : {4,165}*1320
56-fold covers : {56,6}*1344a, {4,42}*1344a, {8,21}*1344, {8,42}*1344a, {4,168}*1344c, {4,168}*1344d, {28,12}*1344b, {28,6}*1344e, {56,6}*1344b, {56,6}*1344c, {28,12}*1344c, {4,84}*1344b, {4,42}*1344b, {4,84}*1344c, {8,42}*1344b, {8,42}*1344c
57-fold covers : {4,171}*1368
58-fold covers : {116,6}*1392b, {4,87}*1392, {4,174}*1392b, {4,174}*1392c
59-fold covers : {4,177}*1416
60-fold covers : {4,180}*1440b, {4,180}*1440c, {8,45}*1440, {20,18}*1440, {4,90}*1440, {24,15}*1440, {12,15}*1440d, {20,3}*1440b, {20,15}*1440b, {60,6}*1440c, {12,30}*1440a, {12,30}*1440b, {60,6}*1440d
61-fold covers : {4,183}*1464
62-fold covers : {124,6}*1488b, {4,93}*1488, {4,186}*1488b, {4,186}*1488c
63-fold covers : {4,189}*1512
65-fold covers : {4,195}*1560
66-fold covers : {44,18}*1584b, {4,99}*1584, {4,198}*1584b, {4,198}*1584c, {12,33}*1584, {12,66}*1584d, {132,6}*1584d
67-fold covers : {4,201}*1608
68-fold covers : {4,204}*1632b, {4,204}*1632c, {8,51}*1632, {68,6}*1632, {4,102}*1632
69-fold covers : {4,207}*1656
70-fold covers : {28,30}*1680b, {20,42}*1680b, {140,6}*1680b, {4,105}*1680, {4,210}*1680b, {4,210}*1680c
71-fold covers : {4,213}*1704
72-fold covers : {4,54}*1728a, {8,27}*1728, {8,54}*1728a, {4,216}*1728c, {4,216}*1728d, {4,108}*1728b, {4,54}*1728b, {4,108}*1728c, {8,54}*1728b, {8,54}*1728c, {72,6}*1728a, {24,9}*1728, {24,18}*1728a, {24,3}*1728, {24,6}*1728a, {36,12}*1728c, {36,6}*1728b, {72,6}*1728b, {72,6}*1728c, {36,12}*1728d, {12,36}*1728e, {12,36}*1728f, {12,18}*1728c, {12,36}*1728g, {12,12}*1728k, {12,12}*1728l, {12,6}*1728b, {12,12}*1728n, {24,18}*1728b, {24,18}*1728c, {24,18}*1728d, {24,6}*1728b, {24,6}*1728c, {24,6}*1728d, {24,18}*1728e, {24,6}*1728e, {12,18}*1728d, {12,36}*1728h, {12,6}*1728f, {12,12}*1728p, {12,9}*1728, {12,36}*1728i, {36,12}*1728i, {12,3}*1728, {12,12}*1728u, {24,6}*1728f, {24,6}*1728g, {12,12}*1728w, {12,6}*1728i, {12,12}*1728y, {4,6}*1728, {4,12}*1728e, {12,12}*1728ab
73-fold covers : {4,219}*1752
74-fold covers : {148,6}*1776b, {4,111}*1776, {4,222}*1776b, {4,222}*1776c
75-fold covers : {4,225}*1800
76-fold covers : {4,228}*1824b, {4,228}*1824c, {8,57}*1824, {76,6}*1824, {4,114}*1824
77-fold covers : {4,231}*1848
78-fold covers : {52,18}*1872b, {4,117}*1872, {4,234}*1872b, {4,234}*1872c, {12,39}*1872, {12,78}*1872d, {156,6}*1872d
79-fold covers : {4,237}*1896
80-fold covers : {40,12}*1920c, {40,12}*1920d, {4,60}*1920b, {4,60}*1920c, {8,15}*1920a, {8,30}*1920a, {8,60}*1920c, {8,60}*1920d, {8,30}*1920b, {8,30}*1920c, {4,240}*1920c, {4,240}*1920d, {40,6}*1920a, {40,12}*1920e, {40,12}*1920f, {40,6}*1920b, {20,6}*1920a, {40,6}*1920c, {20,24}*1920c, {20,24}*1920d, {40,6}*1920d, {20,6}*1920b, {20,12}*1920b, {20,12}*1920c, {40,12}*1920g, {40,12}*1920h, {20,24}*1920e, {20,24}*1920f, {4,60}*1920d, {8,60}*1920e, {8,60}*1920f, {4,30}*1920a, {8,30}*1920d, {8,30}*1920e, {8,30}*1920f, {8,60}*1920g, {8,60}*1920h, {4,120}*1920c, {4,120}*1920d, {8,30}*1920g, {4,60}*1920e, {4,120}*1920e, {4,30}*1920b, {4,120}*1920f, {4,15}*1920b
81-fold covers : {4,243}*1944, {4,9}*1944, {12,9}*1944a, {12,9}*1944b, {12,9}*1944c
82-fold covers : {164,6}*1968b, {4,123}*1968, {4,246}*1968b, {4,246}*1968c
83-fold covers : {4,249}*1992
Permutation Representation (GAP) :
```s0 := (1,2)(3,4);;
s1 := (2,3);;
s2 := (3,4);;
poly := Group([s0,s1,s2]);;

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

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

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

```
References : None.
to this polytope