Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | D3×#423 |
| Fitting | C3×#423 ⋊ C2 |
| Sylow | C3 ⋊ C2×#423 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#423 | C2 |
| prime radical | C23×D3 | C2×C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #423 | D3 |
| hypocenter | C3 | C2×#423 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #423 | |
576 conjugacy classes of subgroups.
| 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)(13,15);(4,5,6,7);(5,7)(8,9,10,13,11,12,14,15);(2,3)84 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1286, #10684969, #10684981, … |
| maximal normal subgroup of | D3×#1286, #10684969, #10684981, … |
| maximal quotient of | D3×#1566, D3×#1567, D3×#1568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11623);