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# Polytope of Type {6,6}

Atlas Canonical Name : {6,6}*96
if this polytope has a name.
Group : SmallGroup(96,226)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 8, 24, 8
Order of s0s1s2 : 4
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{6,6,2} of size 192
{6,6,4} of size 384
{6,6,3} of size 480
{6,6,6} of size 576
{6,6,4} of size 768
{6,6,4} of size 768
{6,6,4} of size 768
{6,6,4} of size 768
{6,6,8} of size 768
{6,6,6} of size 960
{6,6,10} of size 960
{6,6,12} of size 1152
{6,6,14} of size 1344
{6,6,4} of size 1440
{6,6,3} of size 1440
{6,6,18} of size 1728
{6,6,20} of size 1920
{6,6,12} of size 1920
Vertex Figure Of :
{2,6,6} of size 192
{4,6,6} of size 384
{3,6,6} of size 480
{6,6,6} of size 576
{4,6,6} of size 768
{4,6,6} of size 768
{4,6,6} of size 768
{4,6,6} of size 768
{8,6,6} of size 768
{6,6,6} of size 960
{10,6,6} of size 960
{12,6,6} of size 1152
{14,6,6} of size 1344
{4,6,6} of size 1440
{3,6,6} of size 1440
{18,6,6} of size 1728
{20,6,6} of size 1920
{12,6,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6}*48, {6,3}*48
4-fold quotients : {3,3}*24
12-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,12}*192a, {12,6}*192a, {6,12}*192b, {12,6}*192b, {6,6}*192b
3-fold covers : {6,6}*288a, {6,6}*288b
4-fold covers : {12,12}*384a, {12,12}*384b, {6,6}*384c, {6,6}*384d, {6,6}*384e, {6,12}*384, {12,6}*384, {12,12}*384c, {12,12}*384d, {6,24}*384a, {24,6}*384a, {6,24}*384b, {24,6}*384b
5-fold covers : {6,30}*480, {30,6}*480
6-fold covers : {6,12}*576a, {12,6}*576a, {6,12}*576c, {12,6}*576c, {6,6}*576a, {6,6}*576b, {6,12}*576d, {12,6}*576d, {6,12}*576e, {12,6}*576e
7-fold covers : {6,42}*672, {42,6}*672
8-fold covers : {6,12}*768c, {12,6}*768c, {6,12}*768d, {12,6}*768d, {6,12}*768e, {12,6}*768e, {6,6}*768b, {6,6}*768c, {6,6}*768d, {6,24}*768, {24,6}*768, {12,24}*768a, {24,12}*768a, {12,24}*768b, {24,12}*768b, {6,12}*768f, {12,6}*768f, {12,12}*768a, {12,12}*768b, {12,12}*768c, {12,24}*768c, {24,12}*768c, {12,24}*768d, {24,12}*768d, {6,12}*768g, {12,6}*768g, {12,24}*768e, {24,12}*768e, {12,24}*768f, {24,12}*768f, {6,12}*768h, {12,6}*768h, {6,6}*768e, {6,12}*768i, {12,6}*768i, {6,6}*768f, {6,12}*768j, {12,6}*768j, {6,48}*768a, {48,6}*768a, {6,48}*768b, {48,6}*768b
9-fold covers : {6,18}*864, {18,6}*864, {6,6}*864a, {6,6}*864b, {6,6}*864c
10-fold covers : {6,60}*960a, {60,6}*960a, {12,30}*960a, {30,12}*960a, {6,30}*960, {30,6}*960, {6,60}*960b, {60,6}*960b, {12,30}*960b, {30,12}*960b
11-fold covers : {6,66}*1056, {66,6}*1056
12-fold covers : {6,6}*1152a, {6,6}*1152b, {12,12}*1152d, {12,12}*1152e, {12,12}*1152f, {12,12}*1152g, {6,12}*1152a, {12,6}*1152a, {6,6}*1152c, {6,6}*1152d, {6,6}*1152e, {6,6}*1152f, {6,24}*1152g, {24,6}*1152g, {6,24}*1152i, {24,6}*1152i, {12,12}*1152j, {12,12}*1152l, {6,24}*1152j, {24,6}*1152j, {6,12}*1152e, {12,6}*1152e, {12,12}*1152p, {12,12}*1152q, {6,24}*1152m, {24,6}*1152m, {6,12}*1152j, {12,6}*1152j
13-fold covers : {6,78}*1248, {78,6}*1248
14-fold covers : {6,84}*1344a, {84,6}*1344a, {12,42}*1344a, {42,12}*1344a, {6,42}*1344, {42,6}*1344, {6,84}*1344b, {84,6}*1344b, {12,42}*1344b, {42,12}*1344b
15-fold covers : {6,30}*1440g, {30,6}*1440g, {6,30}*1440h, {30,6}*1440h
17-fold covers : {6,102}*1632, {102,6}*1632
18-fold covers : {6,36}*1728a, {36,6}*1728a, {12,18}*1728a, {18,12}*1728a, {6,18}*1728a, {18,6}*1728a, {6,36}*1728c, {36,6}*1728c, {12,18}*1728b, {18,12}*1728b, {6,12}*1728a, {12,6}*1728a, {6,12}*1728c, {12,6}*1728c, {6,6}*1728a, {6,6}*1728b, {6,12}*1728d, {12,6}*1728d, {6,12}*1728e, {12,6}*1728e, {6,12}*1728g, {12,6}*1728g, {6,6}*1728f, {6,12}*1728h, {12,6}*1728h, {6,12}*1728j, {12,6}*1728j, {12,12}*1728z
19-fold covers : {6,114}*1824, {114,6}*1824
20-fold covers : {6,30}*1920a, {30,6}*1920a, {12,60}*1920a, {60,12}*1920a, {12,60}*1920b, {60,12}*1920b, {6,60}*1920, {60,6}*1920, {6,30}*1920b, {30,6}*1920b, {6,30}*1920c, {30,6}*1920c, {6,120}*1920a, {120,6}*1920a, {6,120}*1920b, {120,6}*1920b, {12,60}*1920c, {60,12}*1920c, {24,30}*1920a, {30,24}*1920a, {12,30}*1920, {30,12}*1920, {12,60}*1920d, {60,12}*1920d, {24,30}*1920b, {30,24}*1920b
Permutation Representation (GAP) :
```s0 := ( 8, 9)(11,12)(13,14)(15,16);;
s1 := ( 1, 2)( 3, 5)( 4,11)( 6, 8)( 7,15)( 9,12)(10,13)(14,16);;
s2 := ( 1, 7)( 2,10)( 3, 4)( 5, 6)( 8,13)( 9,14)(11,15)(12,16);;
poly := Group([s0,s1,s2]);;

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

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

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

```
References : None.
to this polytope