Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×D3×#127 |
| Fitting | C2×C3×#127 ⋊ C2 |
| Sylow | C3 ⋊ C22×#127 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×#127 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C2×#127 | D3 |
| hypocenter | C3 | C22×#127 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
3000 conjugacy classes of subgroups; only the 661 normal subgroups are shown.
This group has 3000 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) = 96 |
A minimal faithful permutation representation — 5 generators on 13 points:
(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.
| maximal subgroup of | #10686669, #10710671, C2×D3×#331, … |
| maximal normal subgroup of | #10686669, #10710671, C2×D3×#331, … |
| maximal quotient of | C2×D3×#384, C2×D3×#390, C4×D3×#127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18767);