Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×C8×S4 |
| Fitting | C23×C8 ⋊ D3 |
| derived | A4 ⋊ C22×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | S4 |
| commutator, hypocenter | A4 | C22×C8 |
| Frattini | C4 | C22×S4 |
| Fitting | C23×C8 | D3 |
| socle | C24 | C4×D3 |
| prime radical | C22×S4 | C4 |
| cosocle | C4×A4 | C23 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2×C8 | |
335 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(3,4,6,5);(4,5,6)(7,8,9,11,10,12,13,14);(1,2)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18011);