Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×#395⋊C3 |
| unambiguous semidirect product | (C2×#395)⋊C3 |
| Fitting, Sylow | C2×#395 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #120⋊C3 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C3 |
| Frattini | C23 | C22×A4 |
| Fitting | C2×#395 | C3 |
| socle | C24 | C2×A4 |
| prime radical | C2×C22⋊SL(2,𝔽3) | C2 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C22 | |
193 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 193 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 26 points:
(3,4)(5,8,9,7,6,10)(11,12,15,18)(13,17,21,24,14,23,20,26,19,16,22,25);(1,2)(4,10)(7,8)(11,19,15,21)(12,23,18,25)(13,22,20,14)(16,26,24,17)36 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#395⋊C3, #14302⋊C3, #14601⋊C3, … |
| maximal normal subgroup of | C4×#395⋊C3, #14302⋊C3, #14601⋊C3, … |
| maximal quotient of | C2×#1446⋊C3, C4×#395⋊C3, #14874⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5759);