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# Polytope of Type {4,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,4}*256a
Also Known As : 2T4(2,0)(2,2), {{4,4|2},{4,4}4}. if this polytope has another name.
Group : SmallGroup(256,16888)
Rank : 4
Schlafli Type : {4,4,4}
Number of vertices, edges, etc : 4, 16, 16, 8
Order of s0s1s2s3 : 4
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Locally Toroidal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,4,4,2} of size 512
{4,4,4,3} of size 768
Vertex Figure Of :
{2,4,4,4} of size 512
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4,4}*128, {2,4,4}*128
4-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
8-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
16-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,4,4}*512a, {4,4,8}*512a, {8,4,4}*512b, {8,4,4}*512c, {4,8,4}*512b, {4,8,4}*512d, {4,4,4}*512c, {4,8,4}*512g, {4,8,4}*512h, {4,4,8}*512d
3-fold covers : {12,4,4}*768a, {4,12,4}*768a, {4,4,12}*768b
5-fold covers : {20,4,4}*1280a, {4,20,4}*1280a, {4,4,20}*1280b
7-fold covers : {28,4,4}*1792a, {4,28,4}*1792a, {4,4,28}*1792b
Permutation Representation (GAP) :
```s0 := ( 9,11)(10,12)(13,15)(14,16);;
s1 := ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16);;
s2 := ( 9,13)(10,14)(11,15)(12,16);;
s3 := ( 5, 6)( 7, 8)(13,14)(15,16);;
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,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(16)!( 9,11)(10,12)(13,15)(14,16);
s1 := Sym(16)!( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16);
s2 := Sym(16)!( 9,13)(10,14)(11,15)(12,16);
s3 := Sym(16)!( 5, 6)( 7, 8)(13,14)(15,16);
poly := sub<Sym(16)|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, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;

```
References :
1. Theorem 10C2, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambr\ idge University Press, 2002)

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