Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | SL(2,𝔽3)×H(5,𝔽2) |
| Fitting, Sylow | Q8×H(5,𝔽2) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C24×A4 |
| commutator, cosocle | C2×Q8 | C24×C3 |
| Fitting | Q8×H(5,𝔽2) | C3 |
| hypercenter | C2×H(5,𝔽2) | A4 |
| hypocenter | Q8 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C2×H(5,𝔽2) | |
2121 conjugacy classes of subgroups; only the 513 normal subgroups are shown.
This group has 2121 conjugacy classes of subgroups — too many to draw. Only its 513 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
119 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085165);