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# Polytope of Type {6,6,6,2}

Atlas Canonical Name : {6,6,6,2}*864d
if this polytope has a name.
Group : SmallGroup(864,4704)
Rank : 5
Schlafli Type : {6,6,6,2}
Number of vertices, edges, etc : 6, 18, 18, 6, 2
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,6,6,2,2} of size 1728
Vertex Figure Of :
{2,6,6,6,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,6,6,2}*288a, {6,6,2,2}*288b
6-fold quotients : {6,3,2,2}*144
9-fold quotients : {2,2,6,2}*96, {2,6,2,2}*96
18-fold quotients : {2,2,3,2}*48, {2,3,2,2}*48
27-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,6,12,2}*1728d, {6,12,6,2}*1728c, {6,6,6,4}*1728f, {12,6,6,2}*1728f
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)(10,11)(13,17)(14,16)(15,18)(19,20)(22,26)
(23,25)(24,27)(28,29)(31,35)(32,34)(33,36)(37,38)(40,44)(41,43)(42,45)(46,47)
(49,53)(50,52)(51,54);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)
(16,25)(17,27)(18,26)(28,31)(29,33)(30,32)(35,36)(37,49)(38,51)(39,50)(40,46)
(41,48)(42,47)(43,52)(44,54)(45,53);;
s3 := ( 1,37)( 2,38)( 3,39)( 4,40)( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)(10,28)
(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,46)(20,47)(21,48)
(22,49)(23,50)(24,51)(25,52)(26,53)(27,54);;
s4 := (55,56);;
poly := Group([s0,s1,s2,s3,s4]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(56)!( 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(56)!( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,11)(13,17)(14,16)(15,18)(19,20)
(22,26)(23,25)(24,27)(28,29)(31,35)(32,34)(33,36)(37,38)(40,44)(41,43)(42,45)
(46,47)(49,53)(50,52)(51,54);
s2 := Sym(56)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,22)(11,24)(12,23)(13,19)(14,21)
(15,20)(16,25)(17,27)(18,26)(28,31)(29,33)(30,32)(35,36)(37,49)(38,51)(39,50)
(40,46)(41,48)(42,47)(43,52)(44,54)(45,53);
s3 := Sym(56)!( 1,37)( 2,38)( 3,39)( 4,40)( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)
(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,46)(20,47)
(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54);
s4 := Sym(56)!(55,56);
poly := sub<Sym(56)|s0,s1,s2,s3,s4>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s3*s4*s3*s4, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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 >;

```

to this polytope