Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | A4×#391 |
| unambiguous semidirect product | (C22×#391)⋊C3 |
| Fitting, Sylow | C22×#391 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×A4 |
| commutator | C24 | C22×C3×C4 |
| Fitting | C22×#391 | C3 |
| socle, cosocle | C25 | C23×C3 |
| prime radical | C23×A4 | C23 |
| hypercenter | #391 | A4 |
| hypocenter | C22 | C3×#391 |
| LCS | G ▷ C24 ▷ C22 | |
| UCS | C1 ◁ C23 ◁ #391 | |
1339 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 1339 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084159);