# Polytope of Type {3,2,2,6,12}

Atlas Canonical Name : {3,2,2,6,12}*1728c
if this polytope has a name.
Group : SmallGroup(1728,47409)
Rank : 6
Schlafli Type : {3,2,2,6,12}
Number of vertices, edges, etc : 3, 3, 2, 6, 36, 12
Order of s0s1s2s3s4s5 : 12
Order of s0s1s2s3s4s5s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,2,2,6,6}*864c
3-fold quotients : {3,2,2,6,4}*576a
4-fold quotients : {3,2,2,3,6}*432
6-fold quotients : {3,2,2,6,2}*288
9-fold quotients : {3,2,2,2,4}*192
12-fold quotients : {3,2,2,3,2}*144
18-fold quotients : {3,2,2,2,2}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (2,3);;
s1 := (1,2);;
s2 := (4,5);;
s3 := ( 6,42)( 7,44)( 8,43)( 9,48)(10,50)(11,49)(12,45)(13,47)(14,46)(15,51)
(16,53)(17,52)(18,57)(19,59)(20,58)(21,54)(22,56)(23,55)(24,60)(25,62)(26,61)
(27,66)(28,68)(29,67)(30,63)(31,65)(32,64)(33,69)(34,71)(35,70)(36,75)(37,77)
(38,76)(39,72)(40,74)(41,73);;
s4 := ( 6,64)( 7,63)( 8,65)( 9,61)(10,60)(11,62)(12,67)(13,66)(14,68)(15,73)
(16,72)(17,74)(18,70)(19,69)(20,71)(21,76)(22,75)(23,77)(24,46)(25,45)(26,47)
(27,43)(28,42)(29,44)(30,49)(31,48)(32,50)(33,55)(34,54)(35,56)(36,52)(37,51)
(38,53)(39,58)(40,57)(41,59);;
s5 := ( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(24,33)(25,35)(26,34)(27,36)
(28,38)(29,37)(30,39)(31,41)(32,40)(43,44)(46,47)(49,50)(52,53)(55,56)(58,59)
(60,69)(61,71)(62,70)(63,72)(64,74)(65,73)(66,75)(67,77)(68,76);;
poly := Group([s0,s1,s2,s3,s4,s5]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  s5 := F.6;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5,
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s5*s0*s5,
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5,
s0*s1*s0*s1*s0*s1, s5*s3*s4*s5*s4*s5*s3*s4*s5*s4,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(77)!(2,3);
s1 := Sym(77)!(1,2);
s2 := Sym(77)!(4,5);
s3 := Sym(77)!( 6,42)( 7,44)( 8,43)( 9,48)(10,50)(11,49)(12,45)(13,47)(14,46)
(15,51)(16,53)(17,52)(18,57)(19,59)(20,58)(21,54)(22,56)(23,55)(24,60)(25,62)
(26,61)(27,66)(28,68)(29,67)(30,63)(31,65)(32,64)(33,69)(34,71)(35,70)(36,75)
(37,77)(38,76)(39,72)(40,74)(41,73);
s4 := Sym(77)!( 6,64)( 7,63)( 8,65)( 9,61)(10,60)(11,62)(12,67)(13,66)(14,68)
(15,73)(16,72)(17,74)(18,70)(19,69)(20,71)(21,76)(22,75)(23,77)(24,46)(25,45)
(26,47)(27,43)(28,42)(29,44)(30,49)(31,48)(32,50)(33,55)(34,54)(35,56)(36,52)
(37,51)(38,53)(39,58)(40,57)(41,59);
s5 := Sym(77)!( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(24,33)(25,35)(26,34)
(27,36)(28,38)(29,37)(30,39)(31,41)(32,40)(43,44)(46,47)(49,50)(52,53)(55,56)
(58,59)(60,69)(61,71)(62,70)(63,72)(64,74)(65,73)(66,75)(67,77)(68,76);
poly := sub<Sym(77)|s0,s1,s2,s3,s4,s5>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s5*s5, s0*s2*s0*s2, s1*s2*s1*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s0*s5*s0*s5, s1*s5*s1*s5, s2*s5*s2*s5,
s3*s5*s3*s5, s0*s1*s0*s1*s0*s1, s5*s3*s4*s5*s4*s5*s3*s4*s5*s4,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

```

to this polytope