Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| direct product | D3×SD128 |
| Fitting | C3×SD128 ⋊ C2 |
| Sylow | C3 ⋊ C2×SD128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D32 |
| commutator, cosocle | C3×C32 | C23 |
| Frattini | C32 | C22×D3 |
| Fitting | C3×SD128 | C2 |
| socle | C2×C3 | C2×D32 |
| prime radical | D3×D32 | C2 |
| hypercenter | SD128 | D3 |
| hypocenter | C3 | C2×SD128 |
| LCS | G ▷ C3×C32 ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ SD128 | |
134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#13742 |
| maximal normal subgroup of | D3×#13742 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 90612);