Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #338⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×#1747⋊C3 |
| unambiguous semidirect product | (C2×#1747)⋊C3 |
| Fitting, Sylow | C2×#1747 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | C42⋊C3 |
| commutator, cosocle, hypocenter | #338 | C22×C3 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | C2×#1747 | C3 |
| socle | C22 | #376⋊C3 |
| prime radical | C2×#338⋊C3 | C2 |
| LCS | G ▷ #338 | |
| UCS | C1 ◁ C2×C8 | |
200 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 200 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.
| transitive degree | μt(G) = 64 |
72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083822);