Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | A4×#99 |
| unambiguous semidirect product | (C22×#99)⋊C3 |
| Fitting, Sylow | C22×#99 ⋊ C3 |
272 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(2,3,4)(5,7)(6,9)(8,12)(10,11);(1,4)(2,3)(5,6)(7,9,12,11)(8,10)44 conjugacy classes — view the full character table.
| maximal subgroup of | A4×AΣL(1,𝔽16), A4×#352, A4×#353, … |
| maximal normal subgroup of | A4×AΣL(1,𝔽16), A4×#352, A4×#353, … |
| maximal quotient of | A4×#323, A4×#324, #17414⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5805);