Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×A4×D8 |
| unambiguous semidirect product | (C23×D8)⋊C3 |
| Fitting, Sylow | C23×D8 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×A4 |
| commutator, cosocle | C22×C4 | C23×C3 |
| Frattini | C4 | C23×A4 |
| Fitting | C23×D8 | C3 |
| socle | C24 | C3×D4 |
| hypercenter | C2×D8 | A4 |
| hypocenter | C22 | C2×C3×D8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
456 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(4,5,6)(7,8)(9,14)(10,12)(11,13);(1,2)(3,6)(4,5)(8,11)(9,13)(12,14);(1,2)56 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#357, A4×#361, C4×A4×D8, … |
| maximal normal subgroup of | A4×#357, A4×#361, C4×A4×D8, … |
| maximal quotient of | C2×A4×#102, C2×A4×#107, C4×A4×D8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18206);