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# Polytope of Type {6,6,4}

Atlas Canonical Name : {6,6,4}*288f
if this polytope has a name.
Group : SmallGroup(288,1028)
Rank : 4
Schlafli Type : {6,6,4}
Number of vertices, edges, etc : 6, 18, 12, 4
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,6,4,2} of size 576
Vertex Figure Of :
{2,6,6,4} of size 576
{3,6,6,4} of size 864
{4,6,6,4} of size 1152
{6,6,6,4} of size 1728
{6,6,6,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,3,4}*144
3-fold quotients : {2,6,4}*96b
6-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,6,4}*576b
3-fold covers : {6,18,4}*864e, {6,6,4}*864f, {6,6,12}*864h
4-fold covers : {6,6,8}*1152a, {6,12,4}*1152f, {6,6,4}*1152d, {6,12,4}*1152h, {6,6,8}*1152c, {6,6,8}*1152e, {12,6,4}*1152d, {6,6,4}*1152h, {12,6,4}*1152h
5-fold covers : {6,6,20}*1440d, {6,30,4}*1440f
6-fold covers : {6,18,4}*1728b, {6,6,4}*1728a, {6,6,4}*1728c, {6,6,12}*1728b, {6,6,12}*1728d
Permutation Representation (GAP) :
```s0 := ( 5, 9)( 6,10)( 7,11)( 8,12)(17,21)(18,22)(19,23)(20,24)(29,33)(30,34)
(31,35)(32,36)(41,45)(42,46)(43,47)(44,48)(53,57)(54,58)(55,59)(56,60)(65,69)
(66,70)(67,71)(68,72);;
s1 := ( 1, 5)( 2, 7)( 3, 6)( 4, 8)(10,11)(13,29)(14,31)(15,30)(16,32)(17,25)
(18,27)(19,26)(20,28)(21,33)(22,35)(23,34)(24,36)(37,41)(38,43)(39,42)(40,44)
(46,47)(49,65)(50,67)(51,66)(52,68)(53,61)(54,63)(55,62)(56,64)(57,69)(58,71)
(59,70)(60,72);;
s2 := ( 1,49)( 2,50)( 3,52)( 4,51)( 5,57)( 6,58)( 7,60)( 8,59)( 9,53)(10,54)
(11,56)(12,55)(13,37)(14,38)(15,40)(16,39)(17,45)(18,46)(19,48)(20,47)(21,41)
(22,42)(23,44)(24,43)(25,61)(26,62)(27,64)(28,63)(29,69)(30,70)(31,72)(32,71)
(33,65)(34,66)(35,68)(36,67);;
s3 := ( 1,40)( 2,39)( 3,38)( 4,37)( 5,44)( 6,43)( 7,42)( 8,41)( 9,48)(10,47)
(11,46)(12,45)(13,52)(14,51)(15,50)(16,49)(17,56)(18,55)(19,54)(20,53)(21,60)
(22,59)(23,58)(24,57)(25,64)(26,63)(27,62)(28,61)(29,68)(30,67)(31,66)(32,65)
(33,72)(34,71)(35,70)(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, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s1*s2*s3*s1*s2*s3, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope