Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
| direct product | C2×SL(2,𝔽3)⋊C16 |
| derived | SL(2,𝔽3) ⋊ C2×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C8 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C16 |
| Frattini | C2×C8 | C2×S4 |
| Fitting | C2×C8×Q8 | D3 |
| socle | C23 | A4⋊C8 |
| prime radical | C22×SL(2,𝔽3) | C8 |
| cosocle | C8×SL(2,𝔽3) | C22 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C8 | |
197 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 256 |
128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085728);