Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×#1780⋊C3 |
| unambiguous semidirect product | (C2×#1780)⋊C3 |
| Fitting, Sylow | C2×#1780 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #120⋊C3 |
| commutator, hypocenter | C22×Q8 | C2×C3×C4 |
| Frattini | C22×C4 | C22×A4 |
| Fitting | C2×#1780 | C3 |
| socle | C24 | C4×A4 |
| prime radical | C2×#5948 | C2 |
| cosocle | C22×#41 | C22×C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2×C4 | |
745 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 745 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083835);