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# Polytope of Type {6,6}

Atlas Canonical Name : {6,6}*288b
if this polytope has a name.
Group : SmallGroup(288,1028)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 24, 72, 24
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{6,6,2} of size 576
{6,6,4} of size 1152
{6,6,3} of size 1440
{6,6,6} of size 1728
Vertex Figure Of :
{2,6,6} of size 576
{4,6,6} of size 1152
{4,6,6} of size 1152
{4,6,6} of size 1152
{3,6,6} of size 1440
{6,6,6} of size 1728
{6,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6}*144
3-fold quotients : {6,6}*96
4-fold quotients : {6,6}*72c
6-fold quotients : {3,6}*48, {6,3}*48
8-fold quotients : {3,6}*36
12-fold quotients : {3,3}*24, {6,2}*24
24-fold quotients : {3,2}*12
36-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,6}*576a, {6,12}*576c, {6,6}*576b, {12,6}*576d, {6,12}*576e
3-fold covers : {18,6}*864, {6,6}*864a, {6,6}*864c
4-fold covers : {6,6}*1152a, {12,12}*1152e, {12,12}*1152g, {12,6}*1152a, {6,6}*1152d, {6,6}*1152f, {24,6}*1152g, {24,6}*1152i, {12,12}*1152l, {6,24}*1152j, {6,12}*1152e, {12,12}*1152q, {6,24}*1152m, {6,12}*1152j
5-fold covers : {6,30}*1440g, {30,6}*1440h
6-fold covers : {36,6}*1728a, {18,12}*1728a, {18,6}*1728a, {36,6}*1728c, {18,12}*1728b, {12,6}*1728a, {6,12}*1728c, {6,6}*1728b, {12,6}*1728d, {6,12}*1728e, {6,12}*1728g, {12,6}*1728g, {6,6}*1728f, {6,12}*1728h, {12,6}*1728h
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 5, 9)( 6,11)( 7,10)( 8,12)(13,25)(14,27)(15,26)(16,28)(17,33)
(18,35)(19,34)(20,36)(21,29)(22,31)(23,30)(24,32)(38,39)(41,45)(42,47)(43,46)
(44,48)(49,61)(50,63)(51,62)(52,64)(53,69)(54,71)(55,70)(56,72)(57,65)(58,67)
(59,66)(60,68);;
s1 := ( 1,53)( 2,54)( 3,56)( 4,55)( 5,49)( 6,50)( 7,52)( 8,51)( 9,57)(10,58)
(11,60)(12,59)(13,41)(14,42)(15,44)(16,43)(17,37)(18,38)(19,40)(20,39)(21,45)
(22,46)(23,48)(24,47)(25,65)(26,66)(27,68)(28,67)(29,61)(30,62)(31,64)(32,63)
(33,69)(34,70)(35,72)(36,71);;
s2 := ( 1, 4)( 5, 8)( 9,12)(13,28)(14,26)(15,27)(16,25)(17,32)(18,30)(19,31)
(20,29)(21,36)(22,34)(23,35)(24,33)(37,40)(41,44)(45,48)(49,64)(50,62)(51,63)
(52,61)(53,68)(54,66)(55,67)(56,65)(57,72)(58,70)(59,71)(60,69);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(72)!( 2, 3)( 5, 9)( 6,11)( 7,10)( 8,12)(13,25)(14,27)(15,26)(16,28)
(17,33)(18,35)(19,34)(20,36)(21,29)(22,31)(23,30)(24,32)(38,39)(41,45)(42,47)
(43,46)(44,48)(49,61)(50,63)(51,62)(52,64)(53,69)(54,71)(55,70)(56,72)(57,65)
(58,67)(59,66)(60,68);
s1 := Sym(72)!( 1,53)( 2,54)( 3,56)( 4,55)( 5,49)( 6,50)( 7,52)( 8,51)( 9,57)
(10,58)(11,60)(12,59)(13,41)(14,42)(15,44)(16,43)(17,37)(18,38)(19,40)(20,39)
(21,45)(22,46)(23,48)(24,47)(25,65)(26,66)(27,68)(28,67)(29,61)(30,62)(31,64)
(32,63)(33,69)(34,70)(35,72)(36,71);
s2 := Sym(72)!( 1, 4)( 5, 8)( 9,12)(13,28)(14,26)(15,27)(16,25)(17,32)(18,30)
(19,31)(20,29)(21,36)(22,34)(23,35)(24,33)(37,40)(41,44)(45,48)(49,64)(50,62)
(51,63)(52,61)(53,68)(54,66)(55,67)(56,65)(57,72)(58,70)(59,71)(60,69);
poly := sub<Sym(72)|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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;

```
References : None.
to this polytope