Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C23×#859 |
| derived | SL(2,𝔽3) ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C25 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C23×C4 |
| Frattini | C22 | C23×S4 |
| Fitting | C24×Q8 | D3 |
| prime radical | C24×SL(2,𝔽3) | C2 |
| cosocle | C2×SL(2,𝔽3) | C24 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C25 | |
2653 conjugacy classes of subgroups; only the 559 normal subgroups are shown.
This group has 2653 conjugacy classes of subgroups — too many to draw. Only its 559 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, 1088980);