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# Polytope of Type {6,12}

Atlas Canonical Name : {6,12}*144a
Also Known As : {6,12|2}. if this polytope has another name.
Group : SmallGroup(144,144)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 6, 36, 12
Order of s0s1s2 : 12
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,12,2} of size 288
{6,12,4} of size 576
{6,12,4} of size 576
{6,12,4} of size 576
{6,12,3} of size 576
{6,12,6} of size 864
{6,12,6} of size 864
{6,12,6} of size 864
{6,12,3} of size 864
{6,12,8} of size 1152
{6,12,8} of size 1152
{6,12,4} of size 1152
{6,12,4} of size 1152
{6,12,4} of size 1152
{6,12,6} of size 1152
{6,12,6} of size 1152
{6,12,6} of size 1296
{6,12,6} of size 1296
{6,12,6} of size 1296
{6,12,10} of size 1440
{6,12,12} of size 1728
{6,12,12} of size 1728
{6,12,12} of size 1728
{6,12,3} of size 1728
{6,12,4} of size 1728
{6,12,6} of size 1728
{6,12,6} of size 1728
Vertex Figure Of :
{2,6,12} of size 288
{3,6,12} of size 432
{4,6,12} of size 576
{3,6,12} of size 576
{4,6,12} of size 576
{6,6,12} of size 864
{6,6,12} of size 864
{6,6,12} of size 864
{8,6,12} of size 1152
{4,6,12} of size 1152
{6,6,12} of size 1152
{9,6,12} of size 1296
{3,6,12} of size 1296
{5,6,12} of size 1440
{5,6,12} of size 1440
{10,6,12} of size 1440
{12,6,12} of size 1728
{12,6,12} of size 1728
{12,6,12} of size 1728
{3,6,12} of size 1728
{4,6,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6}*72a
3-fold quotients : {2,12}*48, {6,4}*48a
6-fold quotients : {2,6}*24, {6,2}*24
9-fold quotients : {2,4}*16
12-fold quotients : {2,3}*12, {3,2}*12
18-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,24}*288a, {12,12}*288a
3-fold covers : {6,36}*432a, {18,12}*432a, {6,12}*432b, {6,12}*432g
4-fold covers : {6,48}*576a, {12,12}*576a, {12,24}*576c, {24,12}*576c, {12,24}*576e, {24,12}*576e, {12,12}*576f, {6,12}*576b
5-fold covers : {30,12}*720b, {6,60}*720b
6-fold covers : {6,72}*864a, {18,24}*864a, {6,24}*864b, {12,36}*864a, {36,12}*864a, {12,12}*864c, {6,24}*864f, {12,12}*864h
7-fold covers : {42,12}*1008b, {6,84}*1008b
8-fold covers : {12,24}*1152b, {24,12}*1152b, {24,24}*1152b, {24,24}*1152g, {24,24}*1152i, {24,24}*1152k, {12,48}*1152b, {48,12}*1152b, {12,48}*1152e, {48,12}*1152e, {12,24}*1152e, {24,12}*1152e, {12,12}*1152c, {6,96}*1152c, {24,12}*1152i, {24,12}*1152k, {6,24}*1152d, {12,24}*1152o, {12,24}*1152q, {6,24}*1152h, {6,12}*1152d, {12,12}*1152h, {12,12}*1152k, {12,12}*1152n
9-fold covers : {18,36}*1296a, {18,12}*1296a, {6,36}*1296b, {54,12}*1296a, {6,108}*1296a, {6,12}*1296a, {6,12}*1296b, {18,12}*1296b, {6,36}*1296f, {18,12}*1296c, {6,36}*1296g, {6,36}*1296l, {18,12}*1296l, {6,12}*1296g, {6,12}*1296h, {6,12}*1296i, {6,12}*1296t
10-fold covers : {30,24}*1440b, {6,120}*1440b, {12,60}*1440b, {60,12}*1440b
11-fold covers : {66,12}*1584b, {6,132}*1584b
12-fold covers : {6,144}*1728a, {18,48}*1728a, {6,48}*1728b, {12,36}*1728a, {36,12}*1728a, {12,12}*1728c, {12,72}*1728a, {72,12}*1728a, {24,36}*1728c, {36,24}*1728c, {12,24}*1728d, {24,12}*1728d, {12,72}*1728c, {72,12}*1728c, {24,36}*1728d, {36,24}*1728d, {12,24}*1728f, {24,12}*1728f, {6,48}*1728f, {12,24}*1728o, {24,12}*1728o, {12,24}*1728p, {24,12}*1728p, {12,12}*1728h, {36,12}*1728c, {6,36}*1728b, {12,36}*1728e, {18,12}*1728c, {12,12}*1728l, {6,12}*1728b, {12,12}*1728w, {6,12}*1728h, {6,12}*1728i
13-fold covers : {78,12}*1872b, {6,156}*1872b
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)
(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)(62,63)
(65,66)(68,69)(71,72);;
s1 := ( 1,38)( 2,37)( 3,39)( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)(10,47)
(11,46)(12,48)(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,65)(20,64)(21,66)
(22,71)(23,70)(24,72)(25,68)(26,67)(27,69)(28,56)(29,55)(30,57)(31,62)(32,61)
(33,63)(34,59)(35,58)(36,60);;
s2 := ( 1,58)( 2,59)( 3,60)( 4,55)( 5,56)( 6,57)( 7,61)( 8,62)( 9,63)(10,67)
(11,68)(12,69)(13,64)(14,65)(15,66)(16,70)(17,71)(18,72)(19,40)(20,41)(21,42)
(22,37)(23,38)(24,39)(25,43)(26,44)(27,45)(28,49)(29,50)(30,51)(31,46)(32,47)
(33,48)(34,52)(35,53)(36,54);;
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*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope