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# Polytope of Type {6,6}

Atlas Canonical Name : {6,6}*288a
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
{6,6,4} of size 1152
{6,6,4} of size 1152
{6,6,3} of size 1440
{6,6,6} of size 1728
{6,6,6} of size 1728
Vertex Figure Of :
{2,6,6} of size 576
{4,6,6} of size 1152
{3,6,6} of size 1440
{6,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,3}*144
3-fold quotients : {6,6}*96
4-fold quotients : {6,6}*72b
6-fold quotients : {3,6}*48, {6,3}*48
8-fold quotients : {6,3}*36
12-fold quotients : {3,3}*24, {2,6}*24
24-fold quotients : {2,3}*12
36-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,12}*576a, {12,6}*576c, {6,6}*576a, {6,12}*576d, {12,6}*576e
3-fold covers : {6,18}*864, {6,6}*864b, {6,6}*864c
4-fold covers : {6,6}*1152b, {12,12}*1152d, {12,12}*1152f, {6,12}*1152a, {6,6}*1152c, {6,6}*1152e, {6,24}*1152g, {6,24}*1152i, {12,12}*1152j, {24,6}*1152j, {12,6}*1152e, {12,12}*1152p, {24,6}*1152m, {12,6}*1152j
5-fold covers : {30,6}*1440g, {6,30}*1440h
6-fold covers : {6,36}*1728a, {12,18}*1728a, {6,18}*1728a, {6,36}*1728c, {12,18}*1728b, {6,12}*1728a, {12,6}*1728c, {6,6}*1728a, {6,12}*1728d, {12,6}*1728e, {6,12}*1728g, {12,6}*1728g, {6,6}*1728f, {6,12}*1728h, {12,6}*1728h
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 6, 7)(10,11)(13,25)(14,27)(15,26)(16,28)(17,29)(18,31)(19,30)
(20,32)(21,33)(22,35)(23,34)(24,36)(38,39)(42,43)(46,47)(49,61)(50,63)(51,62)
(52,64)(53,65)(54,67)(55,66)(56,68)(57,69)(58,71)(59,70)(60,72);;
s1 := ( 1,13)( 2,14)( 3,16)( 4,15)( 5,21)( 6,22)( 7,24)( 8,23)( 9,17)(10,18)
(11,20)(12,19)(27,28)(29,33)(30,34)(31,36)(32,35)(37,49)(38,50)(39,52)(40,51)
(41,57)(42,58)(43,60)(44,59)(45,53)(46,54)(47,56)(48,55)(63,64)(65,69)(66,70)
(67,72)(68,71);;
s2 := ( 1,44)( 2,42)( 3,43)( 4,41)( 5,40)( 6,38)( 7,39)( 8,37)( 9,48)(10,46)
(11,47)(12,45)(13,68)(14,66)(15,67)(16,65)(17,64)(18,62)(19,63)(20,61)(21,72)
(22,70)(23,71)(24,69)(25,56)(26,54)(27,55)(28,53)(29,52)(30,50)(31,51)(32,49)
(33,60)(34,58)(35,59)(36,57);;
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*s2*s1*s0*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope