Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | D3×#459 |
| Fitting | C3×#459 ⋊ C2 |
| Sylow | C3 ⋊ C2×#459 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#459 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #459 | D3 |
| hypocenter | C3 | C2×#459 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(6,9)(7,12)(10,11)(13,14,15);(1,2,3,4);(2,4)(5,6)(7,9)(8,11)(10,12);(14,15)57 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1619, D3×#1622, D3×#1625, … |
| maximal normal subgroup of | D3×#1619, D3×#1622, D3×#1625, … |
| maximal quotient of | D3×#1665, D3×#1667, D3×#1669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12906);