Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
| direct product | C23×#1007 |
| Fitting | C24×C3×Q8 ⋊ C2 |
| Sylow | C3 ⋊ C23×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C23×D3 |
| commutator, cosocle | C2×C3 | C27 |
| Frattini | C2 | C26×D3 |
| Fitting | C24×C3×Q8 | C2 |
| socle | C24×C3 | C24 |
| hypercenter | C24×Q8 | D3 |
| hypocenter | C3 | C23×2−1+4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C24×Q8 | |
59806 conjugacy classes of subgroups; only the 32139 normal subgroups are shown.
This group has 59806 conjugacy classes of subgroups — too many to draw. Only its 32139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083458);