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# Polytope of Type {4,6}

Atlas Canonical Name : {4,6}*144
Also Known As : {4,6}4if this polytope has another name.
Group : SmallGroup(144,186)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 12, 36, 18
Order of s0s1s2 : 4
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
{4,6,2} of size 288
{4,6,3} of size 432
{4,6,4} of size 576
{4,6,6} of size 864
{4,6,6} of size 864
{4,6,8} of size 1152
{4,6,9} of size 1296
{4,6,10} of size 1440
{4,6,12} of size 1728
{4,6,12} of size 1728
Vertex Figure Of :
{2,4,6} of size 288
{4,4,6} of size 576
{6,4,6} of size 864
{8,4,6} of size 1152
{8,4,6} of size 1152
{4,4,6} of size 1152
{3,4,6} of size 1296
{4,4,6} of size 1296
{10,4,6} of size 1440
{12,4,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,6}*72
9-fold quotients : {4,2}*16
18-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,6}*288, {4,12}*288
3-fold covers : {4,6}*432a, {12,6}*432e, {12,6}*432f, {4,6}*432b, {12,6}*432h, {12,6}*432i
4-fold covers : {16,6}*576, {4,12}*576, {8,12}*576a, {4,24}*576a, {4,24}*576b, {8,12}*576b
5-fold covers : {4,30}*720, {20,6}*720
6-fold covers : {8,6}*864a, {24,6}*864d, {24,6}*864e, {4,12}*864b, {12,12}*864d, {12,12}*864e, {4,12}*864c, {12,12}*864i, {8,6}*864b, {24,6}*864g, {24,6}*864h, {12,12}*864k
7-fold covers : {4,42}*1008, {28,6}*1008
8-fold covers : {4,24}*1152a, {8,12}*1152a, {8,24}*1152a, {8,24}*1152b, {8,24}*1152c, {8,24}*1152d, {4,48}*1152a, {16,12}*1152a, {4,48}*1152b, {16,12}*1152b, {4,12}*1152a, {8,12}*1152b, {4,24}*1152b, {32,6}*1152
9-fold covers : {4,18}*1296a, {4,18}*1296b, {4,6}*1296a, {12,6}*1296j, {12,6}*1296k, {12,6}*1296l, {12,6}*1296m, {12,6}*1296n, {36,6}*1296m, {12,6}*1296o, {36,6}*1296n, {36,6}*1296o, {12,6}*1296s, {12,6}*1296t, {12,6}*1296u
10-fold covers : {4,60}*1440, {8,30}*1440, {40,6}*1440, {20,12}*1440
11-fold covers : {4,66}*1584, {44,6}*1584
12-fold covers : {16,6}*1728a, {48,6}*1728d, {48,6}*1728e, {4,12}*1728a, {12,12}*1728d, {12,12}*1728e, {8,12}*1728a, {24,12}*1728g, {24,12}*1728h, {4,24}*1728a, {12,24}*1728i, {12,24}*1728j, {4,24}*1728c, {12,24}*1728k, {12,24}*1728l, {8,12}*1728d, {24,12}*1728m, {24,12}*1728n, {4,24}*1728e, {12,24}*1728q, {4,24}*1728g, {12,24}*1728r, {16,6}*1728b, {48,6}*1728g, {8,12}*1728g, {24,12}*1728s, {8,12}*1728h, {24,12}*1728t, {4,12}*1728c, {12,12}*1728r, {48,6}*1728h, {12,12}*1728s, {24,12}*1728u, {12,24}*1728v, {12,24}*1728w, {24,12}*1728x, {4,6}*1728, {12,6}*1728j, {12,12}*1728aa
13-fold covers : {4,78}*1872, {52,6}*1872
Permutation Representation (GAP) :
```s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36)(37,46)(38,47)(39,48)(40,52)(41,53)(42,54)(43,49)(44,50)(45,51)
(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);;
s1 := ( 1,37)( 2,40)( 3,43)( 4,38)( 5,41)( 6,44)( 7,39)( 8,42)( 9,45)(10,46)
(11,49)(12,52)(13,47)(14,50)(15,53)(16,48)(17,51)(18,54)(19,55)(20,58)(21,61)
(22,56)(23,59)(24,62)(25,57)(26,60)(27,63)(28,64)(29,67)(30,70)(31,65)(32,68)
(33,71)(34,66)(35,69)(36,72);;
s2 := ( 1,29)( 2,28)( 3,30)( 4,35)( 5,34)( 6,36)( 7,32)( 8,31)( 9,33)(10,20)
(11,19)(12,21)(13,26)(14,25)(15,27)(16,23)(17,22)(18,24)(37,65)(38,64)(39,66)
(40,71)(41,70)(42,72)(43,68)(44,67)(45,69)(46,56)(47,55)(48,57)(49,62)(50,61)
(51,63)(52,59)(53,58)(54,60);;
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,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope