Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C24 |
| direct product | A4×#95 |
| Fitting, Sylow | C22×#95 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×A4 |
| commutator | C23 | C3×C42 |
| Fitting | C22×#95 | C3 |
| socle, cosocle | C25 | C22×C3 |
| prime radical | C23×A4 | C22 |
| hypercenter | #95 | A4 |
| hypocenter | C22 | C3×#95 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C23 ◁ #95 | |
440 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 440 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 — 2 generators on 16 points:
(5,8,7,6)(10,11,12)(14,16);(1,4,3,2)(9,12)(10,11)(13,14)(15,16)80 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#327, A4×#328, C2×A4×#95, … |
| maximal normal subgroup of | A4×#327, A4×#328, C2×A4×#95, … |
| maximal quotient of | SL(2,𝔽3)×#95, A4×#336, #20336⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5736);