Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | D8×#41⋊C3 |
| unambiguous semidirect product | (D8×#41)⋊C3 |
| Fitting, Sylow | D8×#41 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D4×A4 |
| commutator, cosocle | C4×Q8 | C23×C3 |
| Frattini | C2×C4 | C23×A4 |
| Fitting | D8×#41 | C3 |
| socle | C22 | C2×D4×A4 |
| hypercenter | C4×D8 | A4 |
| hypocenter | Q8 | C2×C3×D8 |
| LCS | G ▷ C4×Q8 ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ C42 ◁ C4×D8 | |
576 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084842);