Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×#5950 |
| Fitting, Sylow | C2×#585 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3 |
| Frattini | C2 | C22×C22⋊A4 |
| Fitting | C2×#585 | C3 |
| socle | C22 | C2×C22⋊A4 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C4 | |
1071 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 1071 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(3,4,7,10)(5,9,17,14,6,15,11,16,12,8,13,18);(1,2)(3,12)(4,15)(5,6)(7,17)(8,9)(10,18)(11,13)(14,16);(5,11)(6,13)(8,14)(9,16)44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20122);