Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#5493 |
| Fitting | C22×C3×C22⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ C2×#534 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C22×C3×C22⋊C4 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C22×C22⋊C4 | D3 |
| hypocenter | C3 | C2×#534 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×C22⋊C4 | |
1984 conjugacy classes of subgroups; only the 565 normal subgroups are shown.
This group has 1984 conjugacy classes of subgroups — too many to draw. Only its 565 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 17 points:
(3,4,5)(6,7,8,9)(11,15)(14,17);(1,2)(10,12)(11,14)(13,16)(15,17);(10,11)(12,14)(13,15)(16,17);(4,5)(7,9)(12,16)(14,17);(1,2)84 conjugacy classes — view the full character table.
| maximal subgroup of | #10686988, #10686991, #10701709, … |
| maximal normal subgroup of | #10686988, #10686991, #10701709, … |
| maximal quotient of | C2×#82121, C2×#82094, C2×#73569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18270);