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# Polytope of Type {2,4,4,2}

Atlas Canonical Name : {2,4,4,2}*128
if this polytope has a name.
Group : SmallGroup(128,2163)
Rank : 5
Schlafli Type : {2,4,4,2}
Number of vertices, edges, etc : 2, 4, 8, 4, 2
Order of s0s1s2s3s4 : 4
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,4,4,2,2} of size 256
{2,4,4,2,3} of size 384
{2,4,4,2,4} of size 512
{2,4,4,2,5} of size 640
{2,4,4,2,6} of size 768
{2,4,4,2,7} of size 896
{2,4,4,2,9} of size 1152
{2,4,4,2,10} of size 1280
{2,4,4,2,11} of size 1408
{2,4,4,2,13} of size 1664
{2,4,4,2,14} of size 1792
{2,4,4,2,15} of size 1920
Vertex Figure Of :
{2,2,4,4,2} of size 256
{3,2,4,4,2} of size 384
{4,2,4,4,2} of size 512
{5,2,4,4,2} of size 640
{6,2,4,4,2} of size 768
{7,2,4,4,2} of size 896
{9,2,4,4,2} of size 1152
{10,2,4,4,2} of size 1280
{11,2,4,4,2} of size 1408
{13,2,4,4,2} of size 1664
{14,2,4,4,2} of size 1792
{15,2,4,4,2} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,2,4,2}*64, {2,4,2,2}*64
4-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,4,4,4}*256, {4,4,4,2}*256, {2,4,8,2}*256a, {2,8,4,2}*256a, {2,4,8,2}*256b, {2,8,4,2}*256b, {2,4,4,2}*256
3-fold covers : {2,4,12,2}*384a, {2,12,4,2}*384a, {2,4,4,6}*384, {6,4,4,2}*384
4-fold covers : {4,4,4,4}*512, {2,4,8,2}*512a, {2,8,4,2}*512a, {2,8,8,2}*512a, {2,8,8,2}*512b, {2,8,8,2}*512c, {2,8,8,2}*512d, {2,4,8,4}*512a, {4,8,4,2}*512a, {2,4,8,4}*512b, {2,4,8,4}*512c, {4,8,4,2}*512b, {4,8,4,2}*512c, {2,4,8,4}*512d, {4,8,4,2}*512d, {2,4,4,8}*512a, {2,8,4,4}*512a, {4,4,8,2}*512a, {8,4,4,2}*512a, {2,4,4,8}*512b, {2,8,4,4}*512b, {4,4,8,2}*512b, {8,4,4,2}*512b, {2,4,4,4}*512a, {2,4,4,4}*512b, {4,4,4,2}*512a, {4,4,4,2}*512b, {2,4,16,2}*512a, {2,16,4,2}*512a, {2,4,16,2}*512b, {2,16,4,2}*512b, {2,4,4,2}*512, {2,4,8,2}*512b, {2,8,4,2}*512b
5-fold covers : {2,4,20,2}*640, {2,20,4,2}*640, {2,4,4,10}*640, {10,4,4,2}*640
6-fold covers : {4,4,4,6}*768, {6,4,4,4}*768, {2,4,4,12}*768, {2,12,4,4}*768, {4,4,12,2}*768, {12,4,4,2}*768, {2,4,12,4}*768a, {4,12,4,2}*768a, {2,4,8,6}*768a, {2,8,4,6}*768a, {6,4,8,2}*768a, {6,8,4,2}*768a, {2,8,12,2}*768a, {2,12,8,2}*768a, {2,4,24,2}*768a, {2,24,4,2}*768a, {2,4,8,6}*768b, {2,8,4,6}*768b, {6,4,8,2}*768b, {6,8,4,2}*768b, {2,8,12,2}*768b, {2,12,8,2}*768b, {2,4,24,2}*768b, {2,24,4,2}*768b, {2,4,4,6}*768a, {6,4,4,2}*768a, {2,4,12,2}*768a, {2,12,4,2}*768a
7-fold covers : {2,4,28,2}*896, {2,28,4,2}*896, {2,4,4,14}*896, {14,4,4,2}*896
9-fold covers : {2,4,4,18}*1152, {18,4,4,2}*1152, {2,4,36,2}*1152a, {2,36,4,2}*1152a, {6,4,4,6}*1152, {2,4,12,6}*1152a, {2,4,12,6}*1152b, {2,12,4,6}*1152, {6,4,12,2}*1152, {6,12,4,2}*1152a, {6,12,4,2}*1152b, {2,4,12,6}*1152c, {6,12,4,2}*1152c, {2,12,12,2}*1152a, {2,12,12,2}*1152b, {2,12,12,2}*1152c, {2,4,4,2}*1152, {2,4,4,6}*1152, {2,4,12,2}*1152, {2,12,4,2}*1152, {6,4,4,2}*1152
10-fold covers : {4,4,4,10}*1280, {10,4,4,4}*1280, {2,4,4,20}*1280, {2,20,4,4}*1280, {4,4,20,2}*1280, {20,4,4,2}*1280, {2,4,20,4}*1280, {4,20,4,2}*1280, {2,4,8,10}*1280a, {2,8,4,10}*1280a, {10,4,8,2}*1280a, {10,8,4,2}*1280a, {2,8,20,2}*1280a, {2,20,8,2}*1280a, {2,4,40,2}*1280a, {2,40,4,2}*1280a, {2,4,8,10}*1280b, {2,8,4,10}*1280b, {10,4,8,2}*1280b, {10,8,4,2}*1280b, {2,8,20,2}*1280b, {2,20,8,2}*1280b, {2,4,40,2}*1280b, {2,40,4,2}*1280b, {2,4,4,10}*1280, {10,4,4,2}*1280, {2,4,20,2}*1280, {2,20,4,2}*1280
11-fold covers : {2,4,4,22}*1408, {22,4,4,2}*1408, {2,4,44,2}*1408, {2,44,4,2}*1408
13-fold covers : {2,4,4,26}*1664, {26,4,4,2}*1664, {2,4,52,2}*1664, {2,52,4,2}*1664
14-fold covers : {4,4,4,14}*1792, {14,4,4,4}*1792, {2,4,4,28}*1792, {2,28,4,4}*1792, {4,4,28,2}*1792, {28,4,4,2}*1792, {2,4,28,4}*1792, {4,28,4,2}*1792, {2,4,8,14}*1792a, {2,8,4,14}*1792a, {14,4,8,2}*1792a, {14,8,4,2}*1792a, {2,8,28,2}*1792a, {2,28,8,2}*1792a, {2,4,56,2}*1792a, {2,56,4,2}*1792a, {2,4,8,14}*1792b, {2,8,4,14}*1792b, {14,4,8,2}*1792b, {14,8,4,2}*1792b, {2,8,28,2}*1792b, {2,28,8,2}*1792b, {2,4,56,2}*1792b, {2,56,4,2}*1792b, {2,4,4,14}*1792, {14,4,4,2}*1792, {2,4,28,2}*1792, {2,28,4,2}*1792
15-fold covers : {2,4,4,30}*1920, {30,4,4,2}*1920, {2,4,60,2}*1920a, {2,60,4,2}*1920a, {6,4,4,10}*1920, {10,4,4,6}*1920, {2,4,12,10}*1920a, {2,12,4,10}*1920, {10,4,12,2}*1920, {10,12,4,2}*1920a, {2,4,20,6}*1920, {2,20,4,6}*1920, {6,4,20,2}*1920, {6,20,4,2}*1920, {2,12,20,2}*1920, {2,20,12,2}*1920
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (4,5)(6,8);;
s2 := ( 3, 4)( 5, 7)( 6, 9)( 8,10);;
s3 := (4,6)(5,8);;
s4 := (11,12);;
poly := Group([s0,s1,s2,s3,s4]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s3*s4*s3*s4, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(12)!(1,2);
s1 := Sym(12)!(4,5)(6,8);
s2 := Sym(12)!( 3, 4)( 5, 7)( 6, 9)( 8,10);
s3 := Sym(12)!(4,6)(5,8);
s4 := Sym(12)!(11,12);
poly := sub<Sym(12)|s0,s1,s2,s3,s4>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3 >;

```

to this polytope