Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
| direct product | C22×#997 |
| Fitting, Sylow | C22×H(5,𝔽2) ⋊ C3 |
| derived | H(5,𝔽2) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3 |
| Frattini | C2 | C22×C22⋊A4 |
| Fitting | C22×H(5,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C23 | |
1225 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 1225 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(3,4)(6,7,10)(9,11,12);(1,2)(5,10)(6,7)(8,12)(9,11);(6,9)(7,11)44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20120);