Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | A4×#562 |
| unambiguous semidirect product | (C22×#562)⋊C3 |
| Fitting, Sylow | C22×#562 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×A4 |
| commutator, socle, cosocle | C24 | C24×C3 |
| Fitting | C22×#562 | C3 |
| hypercenter | #562 | A4 |
| hypocenter | C22 | C3×#562 |
| LCS | G ▷ C24 ▷ C22 | |
| UCS | C1 ◁ C22 ◁ #562 | |
1717 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 1717 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
76 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085033);