Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2 |
| direct product | C16×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C3×C16) |
| unambiguous semidirect product | (D7×C16)⋊C3 |
| Fitting | C7×C16 ⋊ C2×C3 |
| derived, Sylow | C7 ⋊ C2×C3×C16 |
| Hall | D7×C16 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C2×C3×C16 |
| Frattini | C8 | C2×AGL(1,𝔽7) |
| Fitting | C7×C16 | C2×C3 |
| socle | C2×C7 | C2×C3×C8 |
| prime radical | C2×AGL(1,𝔽7) | C8 |
| cosocle | C7×C8 | C22×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C16 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 7);