Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×#5948 |
| Fitting, Sylow | C23×#41 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C3 |
| Frattini | C2 | C22×C22⋊A4 |
| Fitting | C23×#41 | C3 |
| socle | C24 | C2×A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2×C4 | |
1155 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 1155 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(4,5,6)(7,8,11,14)(9,13,17,20,10,19,16,22,15,12,18,21);(1,2)(7,15,11,17)(8,19,14,21)(9,18,16,10)(12,22,20,13);(3,6)(4,5)48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20118);