Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C16 · Schur multiplier (H2) = C210 |
| direct product | C23×D3×C16 |
| Fitting | C23×C3×C16 ⋊ C2 |
| derived, Sylow | C3 ⋊ C24×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C16 | D3 |
| commutator, hypocenter | C3 | C24×C16 |
| Frattini | C8 | C24×D3 |
| Fitting | C23×C3×C16 | C2 |
| socle | C24×C3 | C2×C8 |
| prime radical | C24×D3 | C8 |
| cosocle | C3×C8 | C25 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C23×C16 | |
2590 conjugacy classes of subgroups; only the 1515 normal subgroups are shown.
This group has 2590 conjugacy classes of subgroups — too many to draw. Only its 1515 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
384 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1076025);