Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C24×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C24×C3) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊C25 |
| unambiguous semidirect product | (C2×ΓL(1,𝔽8))⋊C24 |
| Fitting, socle | C24×C7 ⋊ C2×C3 |
| derived, Sylow | C7 ⋊ C25×C3 |
| Hall | C24×D7 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C25 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C7 | C25×C3 |
| Frattini | C1 | G |
| Fitting, socle | C24×C7 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C24 | |
1496 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, 1266);