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# Polytope of Type {12,6}

Atlas Canonical Name : {12,6}*1296u
if this polytope has a name.
Group : SmallGroup(1296,3529)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 108, 324, 54
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {12,6}*432h, {12,6}*432i
6-fold quotients : {12,6}*216c
9-fold quotients : {12,6}*144b, {4,6}*144
18-fold quotients : {4,6}*72, {6,6}*72c
27-fold quotients : {12,2}*48
36-fold quotients : {3,6}*36
54-fold quotients : {6,2}*24
81-fold quotients : {4,2}*16
108-fold quotients : {3,2}*12
162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)
(23,27)(24,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)
(37,64)(38,66)(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)(47,75)
(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);;
s1 := ( 1, 5)( 2, 4)( 3, 6)( 7, 8)(10,32)(11,31)(12,33)(13,29)(14,28)(15,30)
(16,35)(17,34)(18,36)(19,59)(20,58)(21,60)(22,56)(23,55)(24,57)(25,62)(26,61)
(27,63)(37,41)(38,40)(39,42)(43,44)(46,68)(47,67)(48,69)(49,65)(50,64)(51,66)
(52,71)(53,70)(54,72)(73,77)(74,76)(75,78)(79,80);;
s2 := ( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)(22,25)
(23,26)(24,27)(28,64)(29,65)(30,66)(31,70)(32,71)(33,72)(34,67)(35,68)(36,69)
(37,55)(38,56)(39,57)(40,61)(41,62)(42,63)(43,58)(44,59)(45,60)(46,73)(47,74)
(48,75)(49,79)(50,80)(51,81)(52,76)(53,77)(54,78);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)
(22,25)(23,27)(24,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)
(36,59)(37,64)(38,66)(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)
(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);
s1 := Sym(81)!( 1, 5)( 2, 4)( 3, 6)( 7, 8)(10,32)(11,31)(12,33)(13,29)(14,28)
(15,30)(16,35)(17,34)(18,36)(19,59)(20,58)(21,60)(22,56)(23,55)(24,57)(25,62)
(26,61)(27,63)(37,41)(38,40)(39,42)(43,44)(46,68)(47,67)(48,69)(49,65)(50,64)
(51,66)(52,71)(53,70)(54,72)(73,77)(74,76)(75,78)(79,80);
s2 := Sym(81)!( 1,10)( 2,11)( 3,12)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15)
(22,25)(23,26)(24,27)(28,64)(29,65)(30,66)(31,70)(32,71)(33,72)(34,67)(35,68)
(36,69)(37,55)(38,56)(39,57)(40,61)(41,62)(42,63)(43,58)(44,59)(45,60)(46,73)
(47,74)(48,75)(49,79)(50,80)(51,81)(52,76)(53,77)(54,78);
poly := sub<Sym(81)|s0,s1,s2>;

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

```
References : None.
to this polytope