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# Polytope of Type {4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4}*8
Also Known As : square, quadrangle, 2-cube, {4}. if this polytope has another name.
Group : SmallGroup(8,3)
Rank : 2
Schlafli Type : {4}
Number of vertices, edges, etc : 4, 4
Order of s0s1 : 4
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,2} of size 16
{4,3} of size 24
{4,4} of size 32
{4,6} of size 48
{4,3} of size 48
{4,6} of size 48
{4,6} of size 48
{4,8} of size 64
{4,8} of size 64
{4,4} of size 64
{4,9} of size 72
{4,4} of size 72
{4,6} of size 72
{4,10} of size 80
{4,12} of size 96
{4,12} of size 96
{4,12} of size 96
{4,6} of size 96
{4,14} of size 112
{4,5} of size 120
{4,6} of size 120
{4,15} of size 120
{4,8} of size 128
{4,16} of size 128
{4,16} of size 128
{4,4} of size 128
{4,8} of size 128
{4,18} of size 144
{4,9} of size 144
{4,18} of size 144
{4,18} of size 144
{4,4} of size 144
{4,6} of size 144
{4,20} of size 160
{4,5} of size 160
{4,21} of size 168
{4,22} of size 176
{4,24} of size 192
{4,12} of size 192
{4,24} of size 192
{4,6} of size 192
{4,24} of size 192
{4,24} of size 192
{4,12} of size 192
{4,6} of size 192
{4,12} of size 192
{4,4} of size 200
{4,10} of size 200
{4,26} of size 208
{4,27} of size 216
{4,6} of size 216
{4,12} of size 216
{4,28} of size 224
{4,30} of size 240
{4,5} of size 240
{4,6} of size 240
{4,6} of size 240
{4,6} of size 240
{4,10} of size 240
{4,10} of size 240
{4,15} of size 240
{4,30} of size 240
{4,30} of size 240
{4,8} of size 256
{4,16} of size 256
{4,16} of size 256
{4,32} of size 256
{4,32} of size 256
{4,4} of size 256
{4,8} of size 256
{4,8} of size 256
{4,8} of size 256
{4,33} of size 264
{4,34} of size 272
{4,36} of size 288
{4,36} of size 288
{4,36} of size 288
{4,18} of size 288
{4,4} of size 288
{4,12} of size 288
{4,38} of size 304
{4,39} of size 312
{4,40} of size 320
{4,20} of size 320
{4,40} of size 320
{4,5} of size 320
{4,10} of size 320
{4,10} of size 320
{4,42} of size 336
{4,6} of size 336
{4,7} of size 336
{4,8} of size 336
{4,8} of size 336
{4,21} of size 336
{4,42} of size 336
{4,42} of size 336
{4,44} of size 352
{4,45} of size 360
{4,46} of size 368
{4,24} of size 384
{4,48} of size 384
{4,48} of size 384
{4,12} of size 384
{4,24} of size 384
{4,12} of size 384
{4,12} of size 384
{4,48} of size 384
{4,48} of size 384
{4,12} of size 384
{4,6} of size 384
{4,24} of size 384
{4,24} of size 384
{4,12} of size 384
{4,24} of size 384
{4,6} of size 384
{4,24} of size 384
{4,4} of size 392
{4,14} of size 392
{4,50} of size 400
{4,4} of size 400
{4,10} of size 400
{4,51} of size 408
{4,52} of size 416
{4,54} of size 432
{4,27} of size 432
{4,54} of size 432
{4,54} of size 432
{4,6} of size 432
{4,12} of size 432
{4,6} of size 432
{4,12} of size 432
{4,56} of size 448
{4,28} of size 448
{4,56} of size 448
{4,57} of size 456
{4,58} of size 464
{4,60} of size 480
{4,10} of size 480
{4,10} of size 480
{4,12} of size 480
{4,12} of size 480
{4,60} of size 480
{4,60} of size 480
{4,6} of size 480
{4,10} of size 480
{4,15} of size 480
{4,30} of size 480
{4,62} of size 496
{4,63} of size 504
{4,16} of size 512
{4,8} of size 512
{4,16} of size 512
{4,8} of size 512
{4,8} of size 512
{4,16} of size 512
{4,16} of size 512
{4,32} of size 512
{4,32} of size 512
{4,64} of size 512
{4,64} of size 512
{4,4} of size 512
{4,16} of size 512
{4,8} of size 512
{4,16} of size 512
{4,66} of size 528
{4,33} of size 528
{4,66} of size 528
{4,66} of size 528
{4,68} of size 544
{4,69} of size 552
{4,70} of size 560
{4,72} of size 576
{4,36} of size 576
{4,72} of size 576
{4,18} of size 576
{4,72} of size 576
{4,72} of size 576
{4,36} of size 576
{4,18} of size 576
{4,36} of size 576
{4,4} of size 576
{4,12} of size 576
{4,8} of size 576
{4,24} of size 576
{4,8} of size 576
{4,24} of size 576
{4,74} of size 592
{4,75} of size 600
{4,76} of size 608
{4,78} of size 624
{4,39} of size 624
{4,78} of size 624
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{4,40} of size 640
{4,80} of size 640
{4,80} of size 640
{4,20} of size 640
{4,40} of size 640
{4,20} of size 640
{4,20} of size 640
{4,5} of size 640
{4,10} of size 640
{4,20} of size 640
{4,20} of size 640
{4,10} of size 640
{4,81} of size 648
{4,4} of size 648
{4,18} of size 648
{4,9} of size 648
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{4,6} of size 672
{4,6} of size 672
{4,6} of size 672
{4,7} of size 672
{4,8} of size 672
{4,8} of size 672
{4,8} of size 672
{4,8} of size 672
{4,8} of size 672
{4,8} of size 672
{4,14} of size 672
{4,14} of size 672
{4,42} of size 672
{4,86} of size 688
{4,87} of size 696
{4,88} of size 704
{4,44} of size 704
{4,88} of size 704
{4,90} of size 720
{4,45} of size 720
{4,90} of size 720
{4,90} of size 720
{4,5} of size 720
{4,8} of size 720
{4,10} of size 720
{4,6} of size 720
{4,15} of size 720
{4,20} of size 720
{4,30} of size 720
{4,92} of size 736
{4,93} of size 744
{4,94} of size 752
{4,24} of size 768
{4,48} of size 768
{4,48} of size 768
{4,96} of size 768
{4,96} of size 768
{4,12} of size 768
{4,24} of size 768
{4,24} of size 768
{4,24} of size 768
{4,24} of size 768
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{4,24} of size 768
{4,24} of size 768
{4,96} of size 768
{4,96} of size 768
{4,12} of size 768
{4,6} of size 768
{4,12} of size 768
{4,24} of size 768
{4,12} of size 768
{4,24} of size 768
{4,12} of size 768
{4,24} of size 768
{4,12} of size 768
{4,24} of size 768
{4,6} of size 768
{4,6} of size 768
{4,12} of size 768
{4,12} of size 768
{4,48} of size 768
{4,48} of size 768
{4,98} of size 784
{4,4} of size 784
{4,14} of size 784
{4,99} of size 792
{4,100} of size 800
{4,25} of size 800
{4,4} of size 800
{4,20} of size 800
{4,102} of size 816
{4,51} of size 816
{4,102} of size 816
{4,102} of size 816
{4,104} of size 832
{4,52} of size 832
{4,104} of size 832
{4,105} of size 840
{4,106} of size 848
{4,108} of size 864
{4,108} of size 864
{4,108} of size 864
{4,54} of size 864
{4,12} of size 864
{4,12} of size 864
{4,12} of size 864
{4,12} of size 864
{4,110} of size 880
{4,111} of size 888
{4,56} of size 896
{4,112} of size 896
{4,112} of size 896
{4,28} of size 896
{4,56} of size 896
{4,7} of size 896
{4,114} of size 912
{4,57} of size 912
{4,114} of size 912
{4,114} of size 912
{4,116} of size 928
{4,117} of size 936
{4,118} of size 944
{4,120} of size 960
{4,60} of size 960
{4,120} of size 960
{4,20} of size 960
{4,20} of size 960
{4,24} of size 960
{4,24} of size 960
{4,30} of size 960
{4,120} of size 960
{4,120} of size 960
{4,6} of size 960
{4,12} of size 960
{4,20} of size 960
{4,20} of size 960
{4,10} of size 960
{4,12} of size 960
{4,60} of size 960
{4,30} of size 960
{4,60} of size 960
{4,15} of size 960
{4,30} of size 960
{4,30} of size 960
{4,4} of size 968
{4,22} of size 968
{4,122} of size 976
{4,123} of size 984
{4,124} of size 992
{4,10} of size 1000
{4,20} of size 1000
{4,126} of size 1008
{4,63} of size 1008
{4,126} of size 1008
{4,126} of size 1008
{4,28} of size 1008
{4,42} of size 1008
{4,129} of size 1032
{4,130} of size 1040
{4,132} of size 1056
{4,132} of size 1056
{4,132} of size 1056
{4,66} of size 1056
{4,134} of size 1072
{4,135} of size 1080
{4,136} of size 1088
{4,136} of size 1088
{4,68} of size 1088
{4,138} of size 1104
{4,69} of size 1104
{4,138} of size 1104
{4,138} of size 1104
{4,140} of size 1120
{4,35} of size 1120
{4,141} of size 1128
{4,142} of size 1136
{4,72} of size 1152
{4,8} of size 1152
{4,24} of size 1152
{4,144} of size 1152
{4,16} of size 1152
{4,48} of size 1152
{4,144} of size 1152
{4,16} of size 1152
{4,48} of size 1152
{4,36} of size 1152
{4,72} of size 1152
{4,8} of size 1152
{4,12} of size 1152
{4,4} of size 1152
{4,24} of size 1152
{4,36} of size 1152
{4,36} of size 1152
{4,144} of size 1152
{4,144} of size 1152
{4,36} of size 1152
{4,18} of size 1152
{4,72} of size 1152
{4,72} of size 1152
{4,36} of size 1152
{4,72} of size 1152
{4,18} of size 1152
{4,72} of size 1152
{4,6} of size 1152
{4,8} of size 1152
{4,8} of size 1152
{4,12} of size 1152
{4,9} of size 1152
{4,146} of size 1168
{4,147} of size 1176
{4,148} of size 1184
{4,150} of size 1200
{4,75} of size 1200
{4,150} of size 1200
{4,150} of size 1200
{4,5} of size 1200
{4,30} of size 1200
{4,12} of size 1200
{4,30} of size 1200
{4,152} of size 1216
{4,152} of size 1216
{4,76} of size 1216
{4,153} of size 1224
{4,154} of size 1232
{4,156} of size 1248
{4,156} of size 1248
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{4,78} of size 1248
{4,158} of size 1264
{4,159} of size 1272
{4,40} of size 1280
{4,80} of size 1280
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{4,20} of size 1280
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{4,20} of size 1280
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{4,20} of size 1280
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{4,10} of size 1280
{4,162} of size 1296
{4,81} of size 1296
{4,162} of size 1296
{4,162} of size 1296
{4,4} of size 1296
{4,18} of size 1296
{4,18} of size 1296
{4,36} of size 1296
{4,6} of size 1296
{4,12} of size 1296
{4,6} of size 1296
{4,6} of size 1296
{4,9} of size 1296
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{4,18} of size 1296
{4,18} of size 1296
{4,164} of size 1312
{4,5} of size 1320
{4,5} of size 1320
{4,6} of size 1320
{4,10} of size 1320
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{4,11} of size 1320
{4,12} of size 1320
{4,12} of size 1320
{4,165} of size 1320
{4,166} of size 1328
{4,168} of size 1344
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{4,42} of size 1344
{4,168} of size 1344
{4,168} of size 1344
{4,8} of size 1344
{4,8} of size 1344
{4,28} of size 1344
{4,28} of size 1344
{4,8} of size 1344
{4,8} of size 1344
{4,12} of size 1344
{4,12} of size 1344
{4,84} of size 1344
{4,42} of size 1344
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{4,6} of size 1344
{4,8} of size 1344
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{4,14} of size 1344
{4,4} of size 1352
{4,26} of size 1352
{4,170} of size 1360
{4,171} of size 1368
{4,172} of size 1376
{4,174} of size 1392
{4,87} of size 1392
{4,174} of size 1392
{4,174} of size 1392
{4,88} of size 1408
{4,176} of size 1408
{4,176} of size 1408
{4,44} of size 1408
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{4,45} of size 1440
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{4,6} of size 1440
{4,8} of size 1440
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{4,10} of size 1440
{4,10} of size 1440
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{4,28} of size 1568
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{4,20} of size 2000
{4,10} of size 2000
{4,20} of size 2000
Vertex Figure Of :
{2,4} of size 16
{3,4} of size 24
{4,4} of size 32
{6,4} of size 48
{3,4} of size 48
{6,4} of size 48
{6,4} of size 48
{8,4} of size 64
{8,4} of size 64
{4,4} of size 64
{9,4} of size 72
{4,4} of size 72
{6,4} of size 72
{10,4} of size 80
{12,4} of size 96
{12,4} of size 96
{12,4} of size 96
{6,4} of size 96
{14,4} of size 112
{5,4} of size 120
{6,4} of size 120
{15,4} of size 120
{8,4} of size 128
{16,4} of size 128
{16,4} of size 128
{4,4} of size 128
{8,4} of size 128
{18,4} of size 144
{9,4} of size 144
{18,4} of size 144
{18,4} of size 144
{4,4} of size 144
{6,4} of size 144
{20,4} of size 160
{5,4} of size 160
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Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {8}*16
3-fold covers : {12}*24
4-fold covers : {16}*32
5-fold covers : {20}*40
6-fold covers : {24}*48
7-fold covers : {28}*56
8-fold covers : {32}*64
9-fold covers : {36}*72
10-fold covers : {40}*80
11-fold covers : {44}*88
12-fold covers : {48}*96
13-fold covers : {52}*104
14-fold covers : {56}*112
15-fold covers : {60}*120
16-fold covers : {64}*128
17-fold covers : {68}*136
18-fold covers : {72}*144
19-fold covers : {76}*152
20-fold covers : {80}*160
21-fold covers : {84}*168
22-fold covers : {88}*176
23-fold covers : {92}*184
24-fold covers : {96}*192
25-fold covers : {100}*200
26-fold covers : {104}*208
27-fold covers : {108}*216
28-fold covers : {112}*224
29-fold covers : {116}*232
30-fold covers : {120}*240
31-fold covers : {124}*248
32-fold covers : {128}*256
33-fold covers : {132}*264
34-fold covers : {136}*272
35-fold covers : {140}*280
36-fold covers : {144}*288
37-fold covers : {148}*296
38-fold covers : {152}*304
39-fold covers : {156}*312
40-fold covers : {160}*320
41-fold covers : {164}*328
42-fold covers : {168}*336
43-fold covers : {172}*344
44-fold covers : {176}*352
45-fold covers : {180}*360
46-fold covers : {184}*368
47-fold covers : {188}*376
48-fold covers : {192}*384
49-fold covers : {196}*392
50-fold covers : {200}*400
51-fold covers : {204}*408
52-fold covers : {208}*416
53-fold covers : {212}*424
54-fold covers : {216}*432
55-fold covers : {220}*440
56-fold covers : {224}*448
57-fold covers : {228}*456
58-fold covers : {232}*464
59-fold covers : {236}*472
60-fold covers : {240}*480
61-fold covers : {244}*488
62-fold covers : {248}*496
63-fold covers : {252}*504
64-fold covers : {256}*512
65-fold covers : {260}*520
66-fold covers : {264}*528
67-fold covers : {268}*536
68-fold covers : {272}*544
69-fold covers : {276}*552
70-fold covers : {280}*560
71-fold covers : {284}*568
72-fold covers : {288}*576
73-fold covers : {292}*584
74-fold covers : {296}*592
75-fold covers : {300}*600
76-fold covers : {304}*608
77-fold covers : {308}*616
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84-fold covers : {336}*672
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87-fold covers : {348}*696
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89-fold covers : {356}*712
90-fold covers : {360}*720
91-fold covers : {364}*728
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100-fold covers : {400}*800
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103-fold covers : {412}*824
104-fold covers : {416}*832
105-fold covers : {420}*840
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107-fold covers : {428}*856
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109-fold covers : {436}*872
110-fold covers : {440}*880
111-fold covers : {444}*888
112-fold covers : {448}*896
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114-fold covers : {456}*912
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125-fold covers : {500}*1000
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241-fold covers : {964}*1928
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244-fold covers : {976}*1952
245-fold covers : {980}*1960
246-fold covers : {984}*1968
247-fold covers : {988}*1976
248-fold covers : {992}*1984
249-fold covers : {996}*1992
250-fold covers : {1000}*2000
Permutation Representation (GAP) :
```s0 := (2,3);;
s1 := (1,2)(3,4);;
poly := Group([s0,s1]);;

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

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

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

```
References : None.
to this polytope