Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #5450 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#2375⋊C3 |
| unambiguous semidirect product | (C2×#2375)⋊C3 |
| Fitting, Sylow | C2×#2375 ⋊ C3 |
| derived | C24 ⋊ C22×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #5450 |
| commutator, hypocenter | C24 | C22×C3×C4 |
| Frattini | C23 | C23×A4 |
| Fitting | C2×#2375 | C3 |
| socle | C24 | C22×A4 |
| prime radical | C22×#120⋊C3 | C2 |
| cosocle | C25 | C23×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C22 | |
851 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 851 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084207);