# Polytope of Type {12,6}

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

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

```
References : None.
to this polytope