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# Polytope of Type {4,6,6}

Atlas Canonical Name : {4,6,6}*288d
if this polytope has a name.
Group : SmallGroup(288,1028)
Rank : 4
Schlafli Type : {4,6,6}
Number of vertices, edges, etc : 4, 12, 18, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,6,6,2} of size 576
{4,6,6,3} of size 864
{4,6,6,4} of size 1152
{4,6,6,3} of size 1152
{4,6,6,4} of size 1152
{4,6,6,6} of size 1728
{4,6,6,6} of size 1728
{4,6,6,6} of size 1728
Vertex Figure Of :
{2,4,6,6} of size 576
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {4,6,2}*96c
6-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12,6}*576d, {4,12,6}*576f, {4,6,12}*576d, {4,6,6}*576a
3-fold covers : {4,6,18}*864c, {4,18,6}*864c, {4,6,6}*864e, {4,6,6}*864i
4-fold covers : {4,6,6}*1152a, {4,24,6}*1152g, {4,24,6}*1152i, {4,6,24}*1152d, {4,12,12}*1152d, {4,12,12}*1152f, {4,12,6}*1152e, {4,6,12}*1152a, {4,6,6}*1152d, {4,12,6}*1152h, {8,6,6}*1152b, {8,6,6}*1152d, {4,12,6}*1152k, {4,6,12}*1152e
5-fold covers : {4,30,6}*1440d, {4,6,30}*1440e
6-fold covers : {4,12,18}*1728c, {4,12,18}*1728d, {4,6,36}*1728c, {4,36,6}*1728c, {4,36,6}*1728e, {4,18,12}*1728c, {4,12,6}*1728e, {4,12,6}*1728g, {4,6,12}*1728d, {4,6,18}*1728, {4,18,6}*1728a, {4,6,6}*1728b, {4,12,6}*1728l, {4,12,6}*1728m, {4,6,12}*1728j, {4,6,6}*1728c, {12,6,6}*1728a, {12,6,6}*1728b
Permutation Representation (GAP) :
```s0 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)
(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)
(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)
(65,66)(67,68)(69,70)(71,72);;
s1 := ( 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);;
s2 := ( 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);;
s3 := ( 1,41)( 2,42)( 3,43)( 4,44)( 5,37)( 6,38)( 7,39)( 8,40)( 9,45)(10,46)
(11,47)(12,48)(13,53)(14,54)(15,55)(16,56)(17,49)(18,50)(19,51)(20,52)(21,57)
(22,58)(23,59)(24,60)(25,65)(26,66)(27,67)(28,68)(29,61)(30,62)(31,63)(32,64)
(33,69)(34,70)(35,71)(36,72);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s0*s1*s2*s0*s1,
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, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)
(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)
(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)
(63,64)(65,66)(67,68)(69,70)(71,72);
s1 := 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);
s2 := 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);
s3 := Sym(72)!( 1,41)( 2,42)( 3,43)( 4,44)( 5,37)( 6,38)( 7,39)( 8,40)( 9,45)
(10,46)(11,47)(12,48)(13,53)(14,54)(15,55)(16,56)(17,49)(18,50)(19,51)(20,52)
(21,57)(22,58)(23,59)(24,60)(25,65)(26,66)(27,67)(28,68)(29,61)(30,62)(31,63)
(32,64)(33,69)(34,70)(35,71)(36,72);
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, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s2*s1*s0*s1*s2*s0*s1, 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