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# Polytope of Type {6,12}

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

```
References : None.
to this polytope