Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C24 |
| direct product | C2×C16×S4 |
| Fitting | C23×C16 ⋊ D3 |
| derived | A4 ⋊ C22×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C16 | S4 |
| commutator, hypocenter | A4 | C22×C16 |
| Frattini | C8 | C22×S4 |
| Fitting | C23×C16 | D3 |
| socle | C24 | D3×C8 |
| prime radical | C22×S4 | C8 |
| cosocle | C8×A4 | C23 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2×C16 | |
437 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 437 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | C16×#5935, C2×S4×AGL(1,𝔽17), C2×#72319766, … |
| maximal normal subgroup of | S4×#346, S4×#508, S4×#359, … |
| maximal quotient of | C2×S4×AGL(1,𝔽17), S4×M64 |
| automorphism group of | #13737 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088511);