Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#2679⋊C3 |
| unambiguous semidirect product | (C2×#2679)⋊C3 |
| Fitting, Sylow | C2×#2679 ⋊ C3 |
| derived | #2679 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #5464 |
| commutator, cosocle, hypocenter | #2679 | C2×C3 |
| Frattini | C23 | C2×C22⋊A4 |
| Fitting | C2×#2679 | C3 |
| socle | C24 | C22⋊A4 |
| LCS | G ▷ #2679 | |
| UCS | C1 ◁ C22 | |
1222 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084481);