Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C26 |
| direct product | C22×#5948 |
| Fitting, Sylow | C24×#41 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C23×C3 |
| Frattini | C2 | C23×C22⋊A4 |
| Fitting | C24×#41 | C3 |
| socle | C25 | C2×A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C22×C4 | |
11135 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 11135 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085298);