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# Polytope of Type {4,6,6}

Atlas Canonical Name : {4,6,6}*288c
if this polytope has a name.
Group : SmallGroup(288,977)
Rank : 4
Schlafli Type : {4,6,6}
Number of vertices, edges, etc : 4, 12, 18, 6
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,6,6,2} of size 576
{4,6,6,4} of size 1152
{4,6,6,4} of size 1152
{4,6,6,4} of size 1152
{4,6,6,6} of size 1728
{4,6,6,6} of size 1728
Vertex Figure Of :
{2,4,6,6} of size 576
{4,4,6,6} of size 1152
{6,4,6,6} of size 1728
{3,4,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,6,3}*144, {2,6,6}*144b
3-fold quotients : {4,2,6}*96
4-fold quotients : {2,6,3}*72
6-fold quotients : {4,2,3}*48, {2,2,6}*48
9-fold quotients : {4,2,2}*32
12-fold quotients : {2,2,3}*24
18-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,6,12}*576b, {8,6,6}*576c, {4,12,6}*576c
3-fold covers : {4,6,18}*864b, {4,6,6}*864c, {12,6,6}*864e, {4,6,6}*864h
4-fold covers : {4,12,12}*1152c, {4,24,6}*1152a, {8,12,6}*1152c, {4,24,6}*1152d, {8,12,6}*1152f, {4,12,6}*1152c, {8,6,12}*1152c, {4,6,24}*1152c, {16,6,6}*1152c, {4,6,6}*1152f, {4,12,6}*1152j
5-fold covers : {20,6,6}*1440c, {4,30,6}*1440a, {4,6,30}*1440c
6-fold covers : {4,6,36}*1728b, {4,6,12}*1728b, {8,6,18}*1728b, {8,6,6}*1728c, {4,12,18}*1728b, {4,12,6}*1728c, {24,6,6}*1728e, {12,6,12}*1728e, {8,6,6}*1728e, {4,12,6}*1728j, {12,12,6}*1728g, {4,6,12}*1728h
Permutation Representation (GAP) :
```s0 := ( 1,19)( 2,20)( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)
(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)
(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)
(51,69)(52,70)(53,71)(54,72);;
s1 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,34)
(23,35)(24,36)(25,31)(26,32)(27,33)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)
(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)(20,24)
(21,23)(26,27)(28,31)(29,33)(30,32)(35,36)(37,40)(38,42)(39,41)(44,45)(46,49)
(47,51)(48,50)(53,54)(55,58)(56,60)(57,59)(62,63)(64,67)(65,69)(66,68)
(71,72);;
s3 := ( 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)(19,56)(20,55)(21,57)
(22,62)(23,61)(24,63)(25,59)(26,58)(27,60)(28,65)(29,64)(30,66)(31,71)(32,70)
(33,72)(34,68)(35,67)(36,69);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(72)!( 1,19)( 2,20)( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)
(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)
(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)
(50,68)(51,69)(52,70)(53,71)(54,72);
s1 := Sym(72)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)
(22,34)(23,35)(24,36)(25,31)(26,32)(27,33)(40,43)(41,44)(42,45)(49,52)(50,53)
(51,54)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);
s2 := Sym(72)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)
(20,24)(21,23)(26,27)(28,31)(29,33)(30,32)(35,36)(37,40)(38,42)(39,41)(44,45)
(46,49)(47,51)(48,50)(53,54)(55,58)(56,60)(57,59)(62,63)(64,67)(65,69)(66,68)
(71,72);
s3 := 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)(19,56)(20,55)
(21,57)(22,62)(23,61)(24,63)(25,59)(26,58)(27,60)(28,65)(29,64)(30,66)(31,71)
(32,70)(33,72)(34,68)(35,67)(36,69);
poly := sub<Sym(72)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope