Finite Group DatabasebetaFinite Group Databasebeta| order | 1240 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C1 |
| direct product | Q8×ΓL(1,𝔽32) |
| unambiguous semidirect product | (Q8×C31)⋊C5 |
| Fitting, Hall | Q8×C31 ⋊ C5 |
| Sylow | C31 ⋊ C5×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×ΓL(1,𝔽32) |
| commutator, socle, cosocle | C2×C31 | C22×C5 |
| Fitting | Q8×C31 | C5 |
| prime radical | C2×ΓL(1,𝔽32) | C22 |
| hypercenter | Q8 | ΓL(1,𝔽32) |
| hypocenter | C31 | C5×Q8 |
| LCS | G ▷ C2×C31 ▷ C31 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 248 |
55 conjugacy classes — view the full character table.
| maximal subgroup of | #41×ΓL(1,𝔽32), Q8×D31⋊C5, #72277833, … |
| maximal normal subgroup of | #41×ΓL(1,𝔽32), Q8×D31⋊C5, #72277833, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1240, 14);