Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | A4×C4⋊Q8 |
| unambiguous semidirect product | C4⋊(Q8×A4) |
| unambiguous semidirect product | (C22×C4⋊Q8)⋊C3 |
| Fitting, Sylow | C22×C4⋊Q8 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×A4 |
| commutator, socle, cosocle | C24 | C23×C3 |
| Fitting | C22×C4⋊Q8 | C3 |
| prime radical | C22×A4 | C23 |
| hypercenter | C4⋊Q8 | A4 |
| hypocenter | C22 | C3×C4⋊Q8 |
| LCS | G ▷ C24 ▷ C22 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
384 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(2,3,4)(5,8,7,6);(1,4)(2,3)(9,16,12,11)(10,15,14,13);(6,8)(9,14,12,10)(11,13,16,15)56 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#332, A4×#333, A4×#356, … |
| maximal normal subgroup of | A4×#332, A4×#333, A4×#356, … |
| maximal quotient of | A4×#384, A4×#389, A4×#391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18169);