Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C24 |
| direct product | C22×C8×A4 |
| Fitting, Sylow | C24×C8 ⋊ C3 |
| derived | C22 ⋊ C22×C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C8 | A4 |
| commutator, hypocenter | C22 | C22×C3×C8 |
| Frattini | C4 | C23×A4 |
| Fitting | C24×C8 | C3 |
| socle | C25 | C3×C4 |
| prime radical | C23×A4 | C4 |
| cosocle | C22×C4 | C23×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C22×C8 | |
456 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 16 points:
(6,7,8)(9,10,11,13,12,14,15,16);(3,4)(5,8)(6,7);(1,2)128 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#336, A4×#340, A4×#341, … |
| maximal normal subgroup of | A4×#336, A4×#340, A4×#341, … |
| maximal quotient of | C2×C4×C8×A4, C2×A4×C22⋊C8, C2×A4×C4⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18200);