Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of C22⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C16 · Schur multiplier (H2) = C24 |
| direct product | C16×C22⋊A4 |
| Fitting, Sylow | C24×C16 ⋊ C3 |
| derived | C24 ⋊ C3×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | C22⋊A4 |
| commutator, hypocenter | C24 | C3×C16 |
| Frattini | C8 | C2×C22⋊A4 |
| Fitting | C24×C16 | C3 |
| socle | C25 | C3×C8 |
| prime radical | C2×C22⋊A4 | C8 |
| cosocle | C24×C8 | C2×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C16 | |
490 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
128 conjugacy classes — view the full character table.
| maximal subgroup of | C22⋊A4×AGL(1,𝔽17), A42×C16, M32×C22⋊A4, … |
| maximal normal subgroup of | A42×C16, M32×C22⋊A4, C16×#1020 |
| maximal quotient of | C16×#997, C22⋊A4×AGL(1,𝔽17) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084931);