Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C4×C8×A4 |
| unambiguous semidirect product | (C22×C4×C8)⋊C3 |
| Fitting, Sylow | C22×C4×C8 ⋊ C3 |
| derived | C22 ⋊ C3×C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C8 | A4 |
| commutator, hypocenter | C22 | C3×C4×C8 |
| Frattini | C2×C4 | C22×A4 |
| Fitting | C22×C4×C8 | C3 |
| socle | C24 | C2×C3×C4 |
| prime radical | C22×A4 | C2×C4 |
| cosocle | C23×C4 | C22×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C4×C8 | |
200 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 2 generators on 16 points:
(6,7,8)(9,10,11,13,12,14,15,16);(1,2,3,4)(5,8)(6,7)128 conjugacy classes — view the full character table.
| maximal subgroup of | C82×A4, A4×#322, A4×D4⋊C8, … |
| maximal normal subgroup of | C82×A4, A4×#322, A4×D4⋊C8, … |
| maximal quotient of | C82×A4, A4×#322, C8×#131⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5799);