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# Polytope of Type {6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*288a
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 : 4
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
{6,12,4} of size 1152
{6,12,6} of size 1728
{6,12,6} of size 1728
{6,12,6} of size 1728
Vertex Figure Of :
{2,6,12} of size 576
{3,6,12} 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 : {6,12}*144d
3-fold quotients : {6,4}*96
4-fold quotients : {6,6}*72a
6-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
12-fold quotients : {3,4}*24, {2,6}*24, {6,2}*24
24-fold quotients : {2,3}*12, {3,2}*12
36-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,12}*576d, {6,12}*576b, {6,24}*576c, {6,24}*576e, {12,12}*576j
3-fold covers : {6,36}*864, {18,12}*864a, {6,12}*864b, {6,12}*864c
4-fold covers : {6,24}*1152c, {12,24}*1152i, {12,24}*1152k, {6,24}*1152d, {6,12}*1152b, {6,24}*1152e, {24,12}*1152o, {24,12}*1152q, {6,24}*1152h, {6,12}*1152d, {12,12}*1152h, {12,12}*1152k, {12,24}*1152u, {12,24}*1152v, {24,12}*1152w, {24,12}*1152x, {12,12}*1152t
5-fold covers : {30,12}*1440a, {6,60}*1440d
6-fold covers : {12,36}*1728c, {6,36}*1728b, {6,72}*1728b, {6,72}*1728c, {12,36}*1728d, {36,12}*1728e, {18,12}*1728c, {12,12}*1728j, {6,12}*1728b, {18,24}*1728c, {6,24}*1728c, {18,24}*1728e, {6,24}*1728e, {36,12}*1728h, {12,12}*1728o, {6,24}*1728f, {6,24}*1728g, {12,12}*1728v, {6,12}*1728i, {12,12}*1728x
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)( 2,43)( 3,42)( 4,41)( 5,40)( 6,39)( 7,38)( 8,37)( 9,48)(10,47)
(11,46)(12,45)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,60)
(22,59)(23,58)(24,57)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)
(33,72)(34,71)(35,70)(36,69);;
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,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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)( 2,43)( 3,42)( 4,41)( 5,40)( 6,39)( 7,38)( 8,37)( 9,48)
(10,47)(11,46)(12,45)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)
(21,60)(22,59)(23,58)(24,57)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)
(32,61)(33,72)(34,71)(35,70)(36,69);
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,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

```
References : None.
to this polytope