Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
| direct product | C22×C22⋊SL(2,𝔽3) |
| Fitting, Sylow | C24×Q8 ⋊ C3 |
| derived | C22×Q8 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C3 |
| Frattini | C2 | C22×C22⋊A4 |
| Fitting | C24×Q8 | C3 |
| socle | C25 | A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C23 | |
1139 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 1139 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(3,4)(6,7,8)(10,11,13)(14,16,15);(1,2)(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, 20117);