Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×A4×#41 |
| Fitting, Sylow | C23×#41 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, cosocle | C23 | C24×C3 |
| Frattini | C2 | C24×A4 |
| Fitting | C23×#41 | C3 |
| socle | C24 | C23×C3 |
| hypercenter | C2×#41 | A4 |
| hypocenter | C22 | C2×C3×#41 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
1360 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1360 conjugacy classes of subgroups — too many to draw. Only its 210 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 — 4 generators on 14 points:
(1,2)(4,5,6)(7,9,10,13)(8,12,11,14);(3,6)(4,5)(7,8)(9,12)(10,11)(13,14);(8,11)(12,14);(1,2)80 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#408, A4×#410, A4×#413, … |
| maximal normal subgroup of | A4×#408, A4×#410, A4×#413, … |
| maximal quotient of | C2×A4×#117, C2×C4×D4×A4, C2×C4×Q8×A4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20106);