Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of AΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×AΓL(1,𝔽8) |
| unambiguous semidirect product | AGL(1,𝔽8)⋊(C3×C5) |
| unambiguous semidirect product | (C5×AGL(1,𝔽8))⋊C3 |
| Fitting, socle, Hall | C23×C5 ⋊ ΓL(1,𝔽8) |
| derived, Hall | AGL(1,𝔽8) ⋊ C3×C5 |
| Hall | C5×AGL(1,𝔽8) ⋊ C3 |
| Sylow | C23 ⋊ C5×ΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | AΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | AGL(1,𝔽8) | C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C23×C5 | ΓL(1,𝔽8) |
| LCS | G ▷ AGL(1,𝔽8) | |
| UCS | C1 ◁ C5 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 40 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | D5×AΓL(1,𝔽8), C2×C5×AΓL(1,𝔽8), C5×AGL(3,𝔽2), … |
| maximal normal subgroup of | D5×AΓL(1,𝔽8), C2×C5×AΓL(1,𝔽8), C3×C5×AΓL(1,𝔽8) |
| maximal quotient of | C2×C5×AΓL(1,𝔽8), C3×C5×AΓL(1,𝔽8) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 140);