Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C23×#912 |
| Fitting | C23×C3×D8 ⋊ C2 |
| Sylow | C3 ⋊ C23×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | D3×D4 |
| commutator, cosocle | C3×C4 | C26 |
| Frattini | C4 | C25×D3 |
| Fitting | C23×C3×D8 | C2 |
| socle | C24×C3 | C2×D4 |
| hypercenter | C23×D8 | D3 |
| hypocenter | C3 | C23×#135 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C23×C4 ◁ C23×D8 | |
11422 conjugacy classes of subgroups; only the 3659 normal subgroups are shown.
This group has 11422 conjugacy classes of subgroups — too many to draw. Only its 3659 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083352);