Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C23×C4×A4 |
| Fitting, Sylow | C25×C4 ⋊ C3 |
| derived | C22 ⋊ C23×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4 | A4 |
| commutator, hypocenter | C22 | C23×C3×C4 |
| Frattini | C2 | C24×A4 |
| Fitting | C25×C4 | C3 |
| socle | C26 | C2×C3 |
| prime radical | C24×A4 | C2 |
| cosocle | C23 | C24×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C23×C4 | |
2152 conjugacy classes of subgroups; only the 354 normal subgroups are shown.
This group has 2152 conjugacy classes of subgroups — too many to draw. Only its 354 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 — 4 generators on 14 points:
(8,9,10)(11,12,13,14);(5,6)(7,10)(8,9);(3,4);(1,2)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4×#95, C4×A4×C22⋊C4, A4×#380, … |
| maximal normal subgroup of | C2×A4×#95, C4×A4×C22⋊C4, A4×#380, … |
| maximal quotient of | C22×C42×A4, C22×A4×C22⋊C4, C22×A4×C4⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20101);