Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C24×#41⋊C3 |
| Fitting, Sylow | C24×#41 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24×C4 | A4 |
| commutator, cosocle, hypocenter | Q8 | C25×C3 |
| Frattini | C2 | C25×A4 |
| Fitting | C24×#41 | C3 |
| socle | C25 | C2×A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C24×C4 | |
12670 conjugacy classes of subgroups; only the 1429 normal subgroups are shown.
This group has 12670 conjugacy classes of subgroups — too many to draw. Only its 1429 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, 1085297);