Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| direct product | ΓL(1,𝔽8)×SL(2,ℤ/4) |
| unambiguous semidirect product | A4⋊(C4×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C7×SL(2,ℤ/4))⋊C3 |
| derived | C7×A4 ⋊ C3×C4 |
| Sylow | C7 ⋊ C3×SL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | ΓL(1,𝔽8)×S4 |
| commutator, hypocenter | C7×A4 | C3×C4 |
| Fitting, socle | C23×C7 | C3×D3 |
| prime radical | C2×A4×ΓL(1,𝔽8) | C2 |
| cosocle | C2×C7×A4 | C2×C3 |
| LCS | G ▷ C7×A4 | |
| UCS | C1 ◁ C2 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 84 |
50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 520);