Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C25 |
| direct product | D4×C22⋊A4 |
| unambiguous semidirect product | C4⋊(C2×C22⋊A4) |
| Fitting, Sylow | C24×D4 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C22⋊A4 |
| commutator, socle, cosocle | C25 | C22×C3 |
| Fitting | C24×D4 | C3 |
| hypercenter | D4 | C22⋊A4 |
| hypocenter | C24 | C3×D4 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 ◁ D4 | |
1852 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 1852 conjugacy classes of subgroups — too many to draw. Only its 48 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 — 3 generators on 12 points:
(2,4)(6,7,8)(10,11,12);(1,2)(3,4)(9,12)(10,11);(5,8)(6,7)40 conjugacy classes — view the full character table.
| maximal subgroup of | #32878⋊C3, #32881⋊C3, D4×#120⋊C3, … |
| maximal normal subgroup of | #32878⋊C3, #32881⋊C3, D4×#120⋊C3, … |
| maximal quotient of | C22⋊C4×C22⋊A4, C4⋊C4×C22⋊A4, D4×C22⋊SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20115);