Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #120⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#379⋊C3 |
| unambiguous semidirect product | (C2×#379)⋊C3 |
| Fitting, Sylow | C2×#379 ⋊ C3 |
| derived | C24 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #120⋊C3 |
| commutator, hypocenter | C24 | C2×C3×C4 |
| Frattini | C23 | C22×A4 |
| Fitting | C2×#379 | C3 |
| socle | C24 | C2×A4 |
| prime radical | C22×C22⋊A4 | C2 |
| cosocle | C25 | C22×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C22 | |
634 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(3,4,5,6)(7,8)(9,12,13,11,10,14);(1,2)(8,14)(11,12)40 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#379⋊C3, #11709220, #14657⋊C3, … |
| maximal normal subgroup of | C4×#379⋊C3, #11709220, #14657⋊C3, … |
| maximal quotient of | C4×#379⋊C3, C2×#1446⋊C3, #13851⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5744);