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# Polytope of Type {4,2,4}

Atlas Canonical Name : {4,2,4}*64
if this polytope has a name.
Group : SmallGroup(64,226)
Rank : 4
Schlafli Type : {4,2,4}
Number of vertices, edges, etc : 4, 4, 4, 4
Order of s0s1s2s3 : 4
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,2,4,2} of size 128
{4,2,4,3} of size 192
{4,2,4,4} of size 256
{4,2,4,6} of size 384
{4,2,4,3} of size 384
{4,2,4,6} of size 384
{4,2,4,6} of size 384
{4,2,4,9} of size 576
{4,2,4,4} of size 576
{4,2,4,6} of size 576
{4,2,4,10} of size 640
{4,2,4,12} of size 768
{4,2,4,12} of size 768
{4,2,4,12} of size 768
{4,2,4,6} of size 768
{4,2,4,14} of size 896
{4,2,4,5} of size 960
{4,2,4,6} of size 960
{4,2,4,15} of size 960
{4,2,4,18} of size 1152
{4,2,4,4} of size 1152
{4,2,4,6} of size 1152
{4,2,4,9} of size 1152
{4,2,4,18} of size 1152
{4,2,4,18} of size 1152
{4,2,4,20} of size 1280
{4,2,4,5} of size 1280
{4,2,4,21} of size 1344
{4,2,4,22} of size 1408
{4,2,4,4} of size 1600
{4,2,4,10} of size 1600
{4,2,4,26} of size 1664
{4,2,4,27} of size 1728
{4,2,4,6} of size 1728
{4,2,4,12} of size 1728
{4,2,4,28} of size 1792
{4,2,4,30} of size 1920
{4,2,4,15} of size 1920
{4,2,4,30} of size 1920
{4,2,4,30} of size 1920
{4,2,4,5} of size 1920
{4,2,4,6} of size 1920
{4,2,4,6} of size 1920
{4,2,4,6} of size 1920
{4,2,4,10} of size 1920
{4,2,4,10} of size 1920
Vertex Figure Of :
{2,4,2,4} of size 128
{3,4,2,4} of size 192
{4,4,2,4} of size 256
{6,4,2,4} of size 384
{3,4,2,4} of size 384
{6,4,2,4} of size 384
{6,4,2,4} of size 384
{9,4,2,4} of size 576
{4,4,2,4} of size 576
{6,4,2,4} of size 576
{10,4,2,4} of size 640
{12,4,2,4} of size 768
{12,4,2,4} of size 768
{12,4,2,4} of size 768
{6,4,2,4} of size 768
{14,4,2,4} of size 896
{5,4,2,4} of size 960
{6,4,2,4} of size 960
{15,4,2,4} of size 960
{18,4,2,4} of size 1152
{4,4,2,4} of size 1152
{6,4,2,4} of size 1152
{9,4,2,4} of size 1152
{18,4,2,4} of size 1152
{18,4,2,4} of size 1152
{20,4,2,4} of size 1280
{5,4,2,4} of size 1280
{21,4,2,4} of size 1344
{22,4,2,4} of size 1408
{4,4,2,4} of size 1600
{10,4,2,4} of size 1600
{26,4,2,4} of size 1664
{27,4,2,4} of size 1728
{6,4,2,4} of size 1728
{12,4,2,4} of size 1728
{28,4,2,4} of size 1792
{30,4,2,4} of size 1920
{15,4,2,4} of size 1920
{30,4,2,4} of size 1920
{30,4,2,4} of size 1920
{5,4,2,4} of size 1920
{6,4,2,4} of size 1920
{6,4,2,4} of size 1920
{6,4,2,4} of size 1920
{10,4,2,4} of size 1920
{10,4,2,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,2,4}*32, {4,2,2}*32
4-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,4,4}*128, {4,2,8}*128, {8,2,4}*128
3-fold covers : {4,2,12}*192, {12,2,4}*192, {4,6,4}*192a
4-fold covers : {8,2,8}*256, {4,4,8}*256a, {8,4,4}*256a, {4,4,8}*256b, {8,4,4}*256b, {4,8,4}*256a, {4,4,4}*256a, {4,4,4}*256b, {4,8,4}*256b, {4,8,4}*256c, {4,8,4}*256d, {4,2,16}*256, {16,2,4}*256
5-fold covers : {4,2,20}*320, {20,2,4}*320, {4,10,4}*320
6-fold covers : {4,12,4}*384a, {4,4,12}*384, {12,4,4}*384, {4,2,24}*384, {24,2,4}*384, {8,2,12}*384, {12,2,8}*384, {4,6,8}*384a, {8,6,4}*384a
7-fold covers : {4,2,28}*448, {28,2,4}*448, {4,14,4}*448
8-fold covers : {8,4,8}*512a, {8,4,8}*512b, {4,4,4}*512a, {4,8,8}*512a, {8,8,4}*512a, {4,8,8}*512b, {8,8,4}*512b, {4,4,8}*512a, {8,4,4}*512a, {4,8,8}*512c, {8,8,4}*512c, {4,8,8}*512d, {8,8,4}*512d, {4,8,8}*512e, {4,8,8}*512f, {8,8,4}*512e, {8,8,4}*512f, {4,8,8}*512g, {8,8,4}*512g, {4,8,8}*512h, {8,8,4}*512h, {4,4,8}*512b, {8,4,4}*512b, {4,4,8}*512c, {8,4,4}*512c, {4,8,4}*512a, {4,8,4}*512b, {4,8,4}*512c, {4,8,4}*512d, {8,4,8}*512c, {8,4,8}*512d, {4,4,16}*512a, {16,4,4}*512a, {4,4,16}*512b, {16,4,4}*512b, {4,4,4}*512b, {4,4,4}*512c, {4,8,4}*512e, {4,8,4}*512f, {4,8,4}*512g, {4,8,4}*512h, {4,4,8}*512d, {8,4,4}*512d, {4,16,4}*512a, {4,16,4}*512b, {4,16,4}*512c, {4,16,4}*512d
9-fold covers : {4,2,36}*576, {36,2,4}*576, {4,18,4}*576a, {12,2,12}*576, {4,6,12}*576a, {12,6,4}*576a, {4,6,12}*576b, {12,6,4}*576b, {4,6,12}*576c, {12,6,4}*576c, {4,6,4}*576a, {4,6,4}*576b
10-fold covers : {4,20,4}*640, {4,4,20}*640, {20,4,4}*640, {4,2,40}*640, {40,2,4}*640, {8,2,20}*640, {20,2,8}*640, {4,10,8}*640, {8,10,4}*640
11-fold covers : {4,2,44}*704, {44,2,4}*704, {4,22,4}*704
12-fold covers : {8,6,8}*768, {8,2,24}*768, {24,2,8}*768, {8,4,12}*768a, {12,4,8}*768a, {4,12,8}*768a, {8,12,4}*768a, {4,4,24}*768a, {24,4,4}*768a, {8,4,12}*768b, {12,4,8}*768b, {4,12,8}*768b, {8,12,4}*768b, {4,4,24}*768b, {24,4,4}*768b, {4,8,12}*768a, {12,8,4}*768a, {4,24,4}*768a, {4,4,12}*768a, {12,4,4}*768a, {4,12,4}*768a, {4,12,4}*768b, {4,4,12}*768b, {12,4,4}*768b, {4,8,12}*768b, {12,8,4}*768b, {4,24,4}*768b, {4,24,4}*768c, {4,8,12}*768c, {12,8,4}*768c, {4,8,12}*768d, {12,8,4}*768d, {4,24,4}*768d, {4,6,16}*768a, {16,6,4}*768a, {12,2,16}*768, {16,2,12}*768, {4,2,48}*768, {48,2,4}*768, {4,4,12}*768e, {12,4,4}*768e, {4,6,4}*768c, {4,6,4}*768d, {4,6,12}*768a, {12,6,4}*768a
13-fold covers : {4,2,52}*832, {52,2,4}*832, {4,26,4}*832
14-fold covers : {4,28,4}*896, {4,4,28}*896, {28,4,4}*896, {4,2,56}*896, {56,2,4}*896, {8,2,28}*896, {28,2,8}*896, {4,14,8}*896, {8,14,4}*896
15-fold covers : {12,2,20}*960, {20,2,12}*960, {4,6,20}*960a, {20,6,4}*960a, {4,10,12}*960, {12,10,4}*960, {4,2,60}*960, {60,2,4}*960, {4,30,4}*960a
17-fold covers : {4,34,4}*1088, {4,2,68}*1088, {68,2,4}*1088
18-fold covers : {4,4,36}*1152, {36,4,4}*1152, {4,36,4}*1152a, {4,12,12}*1152a, {4,12,12}*1152b, {12,12,4}*1152a, {12,12,4}*1152b, {4,12,12}*1152c, {12,12,4}*1152c, {12,4,12}*1152, {4,4,4}*1152a, {4,4,4}*1152b, {4,12,4}*1152a, {4,12,4}*1152b, {4,4,12}*1152, {12,4,4}*1152, {4,18,8}*1152a, {8,18,4}*1152a, {8,2,36}*1152, {36,2,8}*1152, {4,2,72}*1152, {72,2,4}*1152, {8,6,12}*1152a, {12,6,8}*1152a, {8,6,12}*1152b, {12,6,8}*1152b, {8,6,12}*1152c, {12,6,8}*1152c, {4,6,24}*1152a, {24,6,4}*1152a, {4,6,24}*1152b, {24,6,4}*1152b, {4,6,24}*1152c, {24,6,4}*1152c, {12,2,24}*1152, {24,2,12}*1152, {4,6,8}*1152a, {8,6,4}*1152a, {4,6,8}*1152b, {8,6,4}*1152b
19-fold covers : {4,38,4}*1216, {4,2,76}*1216, {76,2,4}*1216
20-fold covers : {8,10,8}*1280, {8,2,40}*1280, {40,2,8}*1280, {8,4,20}*1280a, {20,4,8}*1280a, {4,20,8}*1280a, {8,20,4}*1280a, {4,4,40}*1280a, {40,4,4}*1280a, {8,4,20}*1280b, {20,4,8}*1280b, {4,20,8}*1280b, {8,20,4}*1280b, {4,4,40}*1280b, {40,4,4}*1280b, {4,8,20}*1280a, {20,8,4}*1280a, {4,40,4}*1280a, {4,4,20}*1280a, {20,4,4}*1280a, {4,20,4}*1280a, {4,20,4}*1280b, {4,4,20}*1280b, {20,4,4}*1280b, {4,8,20}*1280b, {20,8,4}*1280b, {4,40,4}*1280b, {4,40,4}*1280c, {4,8,20}*1280c, {20,8,4}*1280c, {4,8,20}*1280d, {20,8,4}*1280d, {4,40,4}*1280d, {4,10,16}*1280, {16,10,4}*1280, {16,2,20}*1280, {20,2,16}*1280, {4,2,80}*1280, {80,2,4}*1280
21-fold covers : {12,2,28}*1344, {28,2,12}*1344, {4,6,28}*1344a, {28,6,4}*1344a, {4,14,12}*1344, {12,14,4}*1344, {4,2,84}*1344, {84,2,4}*1344, {4,42,4}*1344a
22-fold covers : {4,4,44}*1408, {44,4,4}*1408, {4,44,4}*1408, {4,22,8}*1408, {8,22,4}*1408, {8,2,44}*1408, {44,2,8}*1408, {4,2,88}*1408, {88,2,4}*1408
23-fold covers : {4,46,4}*1472, {4,2,92}*1472, {92,2,4}*1472
25-fold covers : {4,2,100}*1600, {100,2,4}*1600, {4,50,4}*1600, {20,2,20}*1600, {4,10,20}*1600a, {20,10,4}*1600a, {4,10,20}*1600b, {20,10,4}*1600b, {4,10,20}*1600c, {20,10,4}*1600c, {4,10,4}*1600a, {4,10,4}*1600b
26-fold covers : {4,4,52}*1664, {52,4,4}*1664, {4,52,4}*1664, {4,26,8}*1664, {8,26,4}*1664, {8,2,52}*1664, {52,2,8}*1664, {4,2,104}*1664, {104,2,4}*1664
27-fold covers : {4,2,108}*1728, {108,2,4}*1728, {4,54,4}*1728a, {12,2,36}*1728, {36,2,12}*1728, {12,6,12}*1728a, {4,6,36}*1728a, {36,6,4}*1728a, {4,18,12}*1728a, {12,18,4}*1728a, {4,6,12}*1728a, {12,6,4}*1728a, {4,6,36}*1728b, {36,6,4}*1728b, {4,6,12}*1728b, {12,6,4}*1728b, {4,18,12}*1728b, {12,18,4}*1728b, {4,6,12}*1728c, {12,6,4}*1728c, {4,6,4}*1728a, {4,6,4}*1728b, {4,6,12}*1728f, {4,6,12}*1728g, {12,6,4}*1728f, {12,6,4}*1728g, {12,6,12}*1728b, {12,6,12}*1728c, {12,6,12}*1728d, {12,6,12}*1728e, {12,6,12}*1728f, {12,6,12}*1728g, {4,6,12}*1728h, {12,6,4}*1728h, {4,6,12}*1728k, {4,6,12}*1728l, {12,6,4}*1728k, {12,6,4}*1728l, {4,6,4}*1728c, {4,6,4}*1728d, {4,6,12}*1728m, {12,6,4}*1728m, {4,6,12}*1728n, {12,6,4}*1728n
28-fold covers : {8,14,8}*1792, {8,2,56}*1792, {56,2,8}*1792, {8,4,28}*1792a, {28,4,8}*1792a, {4,28,8}*1792a, {8,28,4}*1792a, {4,4,56}*1792a, {56,4,4}*1792a, {8,4,28}*1792b, {28,4,8}*1792b, {4,28,8}*1792b, {8,28,4}*1792b, {4,4,56}*1792b, {56,4,4}*1792b, {4,8,28}*1792a, {28,8,4}*1792a, {4,56,4}*1792a, {4,4,28}*1792a, {28,4,4}*1792a, {4,28,4}*1792a, {4,28,4}*1792b, {4,4,28}*1792b, {28,4,4}*1792b, {4,8,28}*1792b, {28,8,4}*1792b, {4,56,4}*1792b, {4,56,4}*1792c, {4,8,28}*1792c, {28,8,4}*1792c, {4,8,28}*1792d, {28,8,4}*1792d, {4,56,4}*1792d, {4,14,16}*1792, {16,14,4}*1792, {16,2,28}*1792, {28,2,16}*1792, {4,2,112}*1792, {112,2,4}*1792
29-fold covers : {4,58,4}*1856, {4,2,116}*1856, {116,2,4}*1856
30-fold covers : {4,4,60}*1920, {60,4,4}*1920, {4,60,4}*1920a, {4,20,12}*1920, {12,20,4}*1920, {4,12,20}*1920a, {20,12,4}*1920a, {12,4,20}*1920, {20,4,12}*1920, {4,30,8}*1920a, {8,30,4}*1920a, {8,2,60}*1920, {60,2,8}*1920, {4,2,120}*1920, {120,2,4}*1920, {8,10,12}*1920, {12,10,8}*1920, {8,6,20}*1920, {20,6,8}*1920, {4,10,24}*1920, {24,10,4}*1920, {4,6,40}*1920a, {40,6,4}*1920a, {12,2,40}*1920, {40,2,12}*1920, {20,2,24}*1920, {24,2,20}*1920
31-fold covers : {4,62,4}*1984, {4,2,124}*1984, {124,2,4}*1984
Permutation Representation (GAP) :
```s0 := (2,3);;
s1 := (1,2)(3,4);;
s2 := (6,7);;
s3 := (5,6)(7,8);;
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,
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(8)!(2,3);
s1 := Sym(8)!(1,2)(3,4);
s2 := Sym(8)!(6,7);
s3 := Sym(8)!(5,6)(7,8);
poly := sub<Sym(8)|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, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3 >;

```

to this polytope