Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C4×#997 |
| Fitting, Sylow | C4×H(5,𝔽2) ⋊ C3 |
| derived | H(5,𝔽2) ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C22⋊A4 |
| commutator, hypocenter | H(5,𝔽2) | C3×C4 |
| Frattini, socle | C22 | C2×C22⋊A4 |
| Fitting | C4×H(5,𝔽2) | C3 |
| prime radical | C2×#997 | C2 |
| cosocle | C2×H(5,𝔽2) | C2×C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C4 | |
345 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 29 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:
(1,2,3,4)(6,7,9)(10,11,12);(6,10)(7,11);(5,9)(6,7)(8,12)(10,11)44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18220);