Finite Group DatabasebetaFinite Group Databasebeta| order | 1240 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊(D31⋊C5) |
| unambiguous semidirect product | D124⋊C5 |
| Fitting | C4×C31 ⋊ C2×C5 |
| Hall | D124 ⋊ C5 |
| Hall | ΓL(1,𝔽32) ⋊ D4 |
| Sylow | C31 ⋊ C5×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D31⋊C5 |
| commutator, socle, cosocle | C2×C31 | C22×C5 |
| Fitting | C4×C31 | C2×C5 |
| hypercenter | C4 | D31⋊C5 |
| hypocenter | C31 | C5×D4 |
| LCS | G ▷ C2×C31 ▷ C31 | |
| UCS | C1 ◁ C2 ◁ C4 | |
32 conjugacy classes of subgroups.
| transitive degree | μt(G) = 124 |
Polycyclic presentation:
2^2,5,2,31|b^2=d,b^a=bd,e^a=e^30,e^c=e^1637 conjugacy classes — view the full character table.
| maximal subgroup of | D4×D31⋊C5, #72230850, C2×C4⋊(D31⋊C5), … |
| maximal normal subgroup of | D4×D31⋊C5, #72230850, C2×C4⋊(D31⋊C5), … |
| maximal quotient of | C2×C4⋊(D31⋊C5), #72277822, #72277867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1240, 9);