Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×C22⋊(#95⋊C3) |
| Fitting, Sylow | C23×#95 ⋊ C3 |
| derived | C22×#95 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C22⋊(C42⋊C3) |
| commutator, cosocle, hypocenter | C22×#95 | C2×C3 |
| Frattini | C23 | C2×C22⋊A4 |
| Fitting | C23×#95 | C3 |
| socle | C26 | A4 |
| LCS | G ▷ C22×#95 | |
| UCS | C1 ◁ C22 | |
2464 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 2464 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
64 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084115);