Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#5404 |
| Fitting | C4×#131 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C8 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C42 |
| Frattini | C2×C4 | C22×S4 |
| Fitting | C4×#131 | D3 |
| socle | C22 | C2×C4×S4 |
| prime radical | C2×#985 | C22 |
| cosocle | C2×#41⋊C3 | C23 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4×C8 | |
575 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 575 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086963);