Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C22×C4×SL(2,ℤ/4) |
| derived | A4 ⋊ C22×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4 | S4 |
| commutator, hypocenter | A4 | C22×C42 |
| Frattini | C22 | C23×S4 |
| Fitting | C25×C4 | D3 |
| socle | C26 | C2×D3 |
| prime radical | C24×A4 | C22 |
| cosocle | C22×A4 | C24 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C23×C4 | |
3585 conjugacy classes of subgroups; only the 485 normal subgroups are shown.
This group has 3585 conjugacy classes of subgroups — too many to draw. Only its 485 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088664);