Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| isoclinism | isoclinic class of AGL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C72 · Schur multiplier (H2) = C7 |
| direct product | C2×C7×AGL(1,𝔽8) |
| unambiguous semidirect product | C24⋊C72 |
| unambiguous semidirect product | (C24×C7)⋊C7 |
| Fitting, socle | C24×C7 ⋊ C7 |
| derived | C23 ⋊ C2×C72 |
| Sylow | C24 ⋊ C72 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | AGL(1,𝔽8) |
| commutator, cosocle, hypocenter | C23 | C2×C72 |
| Frattini | C1 | G |
| Fitting, socle | C24×C7 | C7 |
| LCS | G ▷ C23 | |
| UCS | C1 ◁ C2×C7 | |
58 conjugacy classes of subgroups.
| transitive degree | μt(G) = 98 |
112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 167);