Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×D4×S4 |
| Fitting | C23×D4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×S4 |
| commutator, cosocle | C2×A4 | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C23×D4 | D3 |
| socle | C24 | C22×D3 |
| hypercenter | C2×D4 | S4 |
| hypocenter | A4 | C22×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
1295 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 1295 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 4 generators on 10 points:
(7,8,10,9);(1,2)(3,4)(5,6)(8,9,10);(1,2)(4,6);(1,2)50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20051);