Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
| direct product | D3×#493 |
| Fitting | C3×#493 ⋊ C2 |
| Sylow | C3 ⋊ C2×#493 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#493 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #493 | D3 |
| hypocenter | C3 | C2×#493 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
760 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3,4)(13,14,15);(5,6,10,12,8,11,7,9);(2,4)(6,9)(7,10)(11,12);(14,15)66 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1357, D3×#1361, #10706439, … |
| maximal normal subgroup of | D3×#1357, D3×#1361, #10706439, … |
| maximal quotient of | D3×#1706, D3×#1709, D3×#1712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14357);