Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C4×C22⋊SL(2,𝔽3) |
| Fitting, Sylow | C22×C4×Q8 ⋊ C3 |
| derived | C22×Q8 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C22⋊A4 |
| commutator, hypocenter | C22×Q8 | C3×C4 |
| Frattini | C22 | C2×C22⋊A4 |
| Fitting | C22×C4×Q8 | C3 |
| socle | C24 | C2×A4 |
| prime radical | C2×C22⋊SL(2,𝔽3) | C2 |
| cosocle | C23×Q8 | C2×C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2×C4 | |
329 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 96 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,4)(6,7,8)(10,11,13)(14,16,15);(9,13,12,15)(10,16,14,11);(5,8)(6,7)48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18219);