Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | A4×#123 |
| unambiguous semidirect product | (C22×#123)⋊C3 |
| Fitting, Sylow | C22×#123 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×A4 |
| commutator, socle, cosocle | C24 | C23×C3 |
| Fitting | C22×#123 | C3 |
| prime radical | C2×D4×A4 | C2 |
| hypercenter | #123 | A4 |
| hypocenter | C22 | C3×#123 |
| LCS | G ▷ C24 ▷ C22 | |
| UCS | C1 ◁ C22 ◁ #123 | |
458 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,3,4)(6,8)(9,15,12,11)(10,16,13,14);(1,4)(2,3)(10,13)(14,16);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)56 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#352, A4×#458, C2×A4×#123, … |
| maximal normal subgroup of | A4×#352, A4×#458, C2×A4×#123, … |
| maximal quotient of | A4×#381, A4×#385, A4×#386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18162);