Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | A4×H(5,𝔽2) |
| Fitting, Sylow | C22×H(5,𝔽2) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×A4 |
| commutator, socle, cosocle | C23 | C24×C3 |
| Fitting | C22×H(5,𝔽2) | C3 |
| hypercenter | H(5,𝔽2) | A4 |
| hypocenter | C22 | C3×H(5,𝔽2) |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
1370 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 1370 conjugacy classes of subgroups — too many to draw. Only its 204 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) = 24 |
A minimal faithful permutation representation — 4 generators on 12 points:
(2,3,4)(6,10)(9,11);(1,4)(2,3)(7,12)(9,11);(5,7)(6,9)(8,12)(10,11);(5,6)(7,11)(8,10)(9,12)68 conjugacy classes — view the full character table.
| derived subgroup of | S4×#997 |
| maximal subgroup of | A4×#453, A4×#455, A4×B(4,𝔽2), … |
| maximal normal subgroup of | A4×#453, A4×#455, A4×B(4,𝔽2), … |
| maximal quotient of | A4×#518, A4×#520, A4×#534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20108);