Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C2×D3×#120 |
| Fitting | C2×C3×#120 ⋊ C2 |
| Sylow | C3 ⋊ C22×#120 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×#120 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C2×#120 | D3 |
| hypocenter | C3 | C22×#120 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
4544 conjugacy classes of subgroups; only the 661 normal subgroups are shown.
This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 661 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 5 generators on 13 points:
(1,2)(3,4,5)(6,7)(8,9);(10,11)(12,13);(1,2)(7,9)(11,13);(4,5);(1,2)84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18491);