Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C26×C3 |
| direct product | C23×C3×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C23×C32) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C24×C3) |
| unambiguous semidirect product | (C2×ΓL(1,𝔽8))⋊(C23×C3) |
| Fitting, socle | C23×C3×C7 ⋊ C2×C3 |
| derived, Sylow | C7 ⋊ C24×C32 |
| Hall | C23×D7 ⋊ C32 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C7 | C24×C32 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3×C7 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C23×C3 | |
804 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 928);