Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×D4×#67 |
| Fitting | C23×C3×D4 ⋊ C2 |
| Sylow | C3 ⋊ C2×D42 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C23×C3×D4 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C23×D4 | D3 |
| hypocenter | C3 | C2×D42 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×D4 | |
2536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 13 points:
(1,2)(3,4,5)(7,9);(4,5)(6,7)(8,9);(1,2)(10,11)(12,13);(11,13);(1,2)90 conjugacy classes — view the full character table.
| maximal subgroup of | #10686849, #10810505, D4×#806, … |
| maximal normal subgroup of | #10686849, #10810505, D4×#806, … |
| maximal quotient of | C2×D4×(C2×D3)⋊C4, C2×C22⋊C4×#67, C2×D4×#217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19814);