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# Polytope of Type {2,6,4}

Atlas Canonical Name : {2,6,4}*96c
if this polytope has a name.
Group : SmallGroup(96,226)
Rank : 4
Schlafli Type : {2,6,4}
Number of vertices, edges, etc : 2, 6, 12, 4
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,6,4,2} of size 192
{2,6,4,4} of size 768
Vertex Figure Of :
{2,2,6,4} of size 192
{3,2,6,4} of size 288
{4,2,6,4} of size 384
{5,2,6,4} of size 480
{6,2,6,4} of size 576
{7,2,6,4} of size 672
{8,2,6,4} of size 768
{9,2,6,4} of size 864
{10,2,6,4} of size 960
{11,2,6,4} of size 1056
{12,2,6,4} of size 1152
{13,2,6,4} of size 1248
{14,2,6,4} of size 1344
{15,2,6,4} of size 1440
{17,2,6,4} of size 1632
{18,2,6,4} of size 1728
{19,2,6,4} of size 1824
{20,2,6,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,12,4}*192b, {2,12,4}*192c, {4,6,4}*192b, {2,6,4}*192
3-fold covers : {2,18,4}*288b, {6,6,4}*288d, {6,6,4}*288e
4-fold covers : {4,12,4}*384b, {4,12,4}*384c, {2,6,4}*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, {4,6,4}*384d
5-fold covers : {10,6,4}*480b, {2,30,4}*480b
6-fold covers : {2,36,4}*576b, {2,36,4}*576c, {4,18,4}*576b, {2,18,4}*576, {6,12,4}*576d, {6,12,4}*576e, {6,12,4}*576f, {6,12,4}*576g, {12,6,4}*576d, {12,6,4}*576e, {6,6,4}*576a, {6,6,4}*576b, {2,6,12}*576a, {2,6,12}*576b
7-fold covers : {14,6,4}*672b, {2,42,4}*672b
8-fold covers : {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, {4,6,4}*768h, {4,12,4}*768j, {4,12,4}*768k, {4,12,4}*768o, {4,12,4}*768p, {8,6,4}*768e, {8,6,4}*768g, {4,6,4}*768l
9-fold covers : {2,54,4}*864b, {18,6,4}*864c, {6,18,4}*864c, {6,18,4}*864d, {6,6,4}*864e, {6,6,4}*864g, {6,6,4}*864i
10-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, {10,6,4}*960e, {2,6,20}*960c, {2,30,4}*960
11-fold covers : {22,6,4}*1056b, {2,66,4}*1056b
12-fold covers : {4,36,4}*1152b, {4,36,4}*1152c, {2,18,4}*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, {6,6,4}*1152a, {6,6,4}*1152b, {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, {4,18,4}*1152d, {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, {6,6,4}*1152g, {6,12,4}*1152l, {12,6,4}*1152f, {12,6,4}*1152g
13-fold covers : {26,6,4}*1248b, {2,78,4}*1248b
14-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, {14,6,4}*1344, {2,6,28}*1344, {2,42,4}*1344
15-fold covers : {10,18,4}*1440b, {2,90,4}*1440b, {30,6,4}*1440d, {6,30,4}*1440d, {6,30,4}*1440e, {30,6,4}*1440e
17-fold covers : {34,6,4}*1632b, {2,102,4}*1632b
18-fold covers : {2,108,4}*1728b, {2,108,4}*1728c, {4,54,4}*1728b, {2,54,4}*1728, {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, {12,18,4}*1728d, {12,6,4}*1728e, {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, {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
19-fold covers : {38,6,4}*1824b, {2,114,4}*1824b
20-fold covers : {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, {2,120,4}*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, {20,6,4}*1920d, {4,30,4}*1920d
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (3,6)(4,8);;
s2 := (3,4)(5,6)(7,8);;
s3 := (3,4)(6,8);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```

to this polytope