Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | D4×#213 |
| Fitting | C3×D42 ⋊ C2 |
| Sylow | C3 ⋊ D4×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×#67 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×D42 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | D42 | D3 |
| hypocenter | C3 | D4×D8 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ D42 | |
620 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 149 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)(9,12)(10,15)(13,14);(2,3)(8,9)(10,12)(11,14)(13,15);(5,7);(4,5)(6,7)60 conjugacy classes — view the full character table.
| maximal subgroup of | #10784527, #10784528, #10784530, … |
| maximal normal subgroup of | #10784527, #10784528, #10784530, … |
| maximal quotient of | #10784528, #10784534, #10784543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16066);