Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #67607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#67607 |
| Fitting | C2×C3×D32 ⋊ C2 |
| Sylow | C3 ⋊ C2×D64 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #4519 |
| commutator, cosocle | C3×C32 | C23 |
| Frattini | C32 | C22×D3 |
| Fitting | C2×C3×D32 | C2 |
| socle | C22×C3 | D32 |
| hypercenter | C2×D32 | D3 |
| hypocenter | C3 | C2×D64 |
| LCS | G ▷ C3×C32 ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ C2×D32 | |
142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 90610);