Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C42×A4 |
| Fitting, Sylow | C23×C42 ⋊ C3 |
| derived | C22 ⋊ C2×C3×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C42 | A4 |
| commutator, hypocenter | C22 | C2×C3×C42 |
| Frattini | C22 | C23×A4 |
| Fitting | C23×C42 | C3 |
| socle | C25 | C22×C3 |
| prime radical | C23×A4 | C22 |
| cosocle | C24 | C23×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C2×C42 | |
680 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 96 |
A minimal faithful permutation representation — 3 generators on 14 points:
(4,5,6)(7,8,9,10);(3,6)(4,5)(11,12,13,14);(1,2)128 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#336, A4×#339, C43×A4, … |
| maximal normal subgroup of | A4×#336, A4×#339, C43×A4, … |
| maximal quotient of | C43×A4, C2×A4×#95, A4×#376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18145);