Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2 |
| direct product | C2×C16×SL(2,𝔽3) |
| unambiguous semidirect product | (C2×Q8×C16)⋊C3 |
| Fitting, Sylow | C2×Q8×C16 ⋊ C3 |
| derived | Q8 ⋊ C2×C3×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C16 | A4 |
| commutator, hypocenter | Q8 | C2×C3×C16 |
| Frattini | C2×C8 | C22×A4 |
| Fitting | C2×Q8×C16 | C3 |
| socle | C23 | C8×A4 |
| prime radical | C22×SL(2,𝔽3) | C8 |
| cosocle | C8×Q8 | C22×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C16 | |
202 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 256 |
224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084079);