Finite Group DatabasebetaFinite Group Databasebeta| order | 11664 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9 · Schur multiplier (H2) = C22×C32 |
No aliases for this group.
| Sylow | C24 ⋊ C3×#6875 |
699 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 699 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,6,3,5,12,9,14,15,4,7,17,8,13,25,20,11,22,10,18,16,19,26,24,21,23,27)(28,31)(29,30)(32,35,33);(29,31,30)(33,35,34);(2,5,11,7,13,23,18,26,14)(3,8,19)(6,15,12,16,27,24,17,22,25)(9,21,20)Polycyclic presentation:
3^6,2^4|a^3=d,c^3=e,e^3=f^2,c^a=ce,d^c=df,e^a=ef^2,g^b=h,g^c=h,h^b=gh,h^c=gh,i^b=j,i^c=ghi,j^b=ij,j^c=gj208 conjugacy classes — view the full character table.
| maximal subgroup of | #72259339, C2×#72229286 |
| maximal normal subgroup of | #72259339, C2×#72229286 |
| maximal quotient of | C2×#72229286, #72276206 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [3, 3, 3, 3, 3, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6^2);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.8);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.9*F.10);;
gap> SetConjugate(coll, 10, 3, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);