Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
| direct product | D3×2−1+6 |
| Fitting | C3×2−1+6 ⋊ C2 |
| Sylow | C3 ⋊ C2×2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C26×D3 |
| commutator, socle, cosocle | C2×C3 | C27 |
| Fitting | C3×2−1+6 | C2 |
| hypercenter | 2−1+6 | D3 |
| hypocenter | C3 | C2×2−1+6 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ 2−1+6 | |
59186 conjugacy classes of subgroups; only the 32041 normal subgroups are shown.
This group has 59186 conjugacy classes of subgroups — too many to draw. Only its 32041 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
195 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083466);