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# Polytope of Type {6,6,6}

Atlas Canonical Name : {6,6,6}*432f
if this polytope has a name.
Group : SmallGroup(432,759)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 6, 18, 18, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,6,6,2} of size 864
{6,6,6,3} of size 1296
{6,6,6,4} of size 1728
Vertex Figure Of :
{2,6,6,6} of size 864
{3,6,6,6} of size 1296
{4,6,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,3,6}*216
3-fold quotients : {2,6,6}*144c, {6,6,2}*144b
6-fold quotients : {2,3,6}*72, {6,3,2}*72
9-fold quotients : {2,6,2}*48
18-fold quotients : {2,3,2}*24
27-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,12,6}*864e, {6,6,12}*864g, {12,6,6}*864g
3-fold covers : {6,18,6}*1296d, {6,6,6}*1296e, {6,6,6}*1296l, {6,6,6}*1296r, {6,6,6}*1296t
4-fold covers : {6,24,6}*1728e, {6,12,12}*1728f, {12,12,6}*1728e, {6,6,24}*1728g, {24,6,6}*1728g, {12,6,12}*1728g, {6,6,6}*1728c, {6,6,6}*1728e, {6,6,12}*1728d, {12,6,6}*1728d
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)
(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54);;
s1 := ( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,20)(11,19)(12,21)(13,26)(14,25)(15,27)
(16,23)(17,22)(18,24)(28,29)(31,35)(32,34)(33,36)(37,47)(38,46)(39,48)(40,53)
(41,52)(42,54)(43,50)(44,49)(45,51);;
s2 := ( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,43)( 8,45)( 9,44)(10,31)
(11,33)(12,32)(13,28)(14,30)(15,29)(16,34)(17,36)(18,35)(19,49)(20,51)(21,50)
(22,46)(23,48)(24,47)(25,52)(26,54)(27,53);;
s3 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54);;
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*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(54)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)
(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54);
s1 := Sym(54)!( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,20)(11,19)(12,21)(13,26)(14,25)
(15,27)(16,23)(17,22)(18,24)(28,29)(31,35)(32,34)(33,36)(37,47)(38,46)(39,48)
(40,53)(41,52)(42,54)(43,50)(44,49)(45,51);
s2 := Sym(54)!( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,43)( 8,45)( 9,44)
(10,31)(11,33)(12,32)(13,28)(14,30)(15,29)(16,34)(17,36)(18,35)(19,49)(20,51)
(21,50)(22,46)(23,48)(24,47)(25,52)(26,54)(27,53);
s3 := Sym(54)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54);
poly := sub<Sym(54)|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*s0*s1*s0*s1*s2*s0*s1*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2 >;

```
References : None.
to this polytope