Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of C42⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C2×C4×C42⋊C3 |
| unambiguous semidirect product | (C2×C43)⋊C3 |
| Fitting, Sylow | C2×C43 ⋊ C3 |
| derived | C42 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C42⋊C3 |
| commutator, hypocenter | C42 | C2×C3×C4 |
| Frattini | C23 | C22×A4 |
| Fitting | C2×C43 | C3 |
| socle | C24 | C2×A4 |
| prime radical | C22×C42⋊C3 | C2 |
| cosocle | C2×C42 | C22×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C2×C4 | |
252 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(3,8,13,7,6,12,5,11,10,4,9,14);(1,2)(6,8,9,11)(10,14,13,12)64 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5734);