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# Polytope of Type {6,4,3}

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

```
References : None.
to this polytope