Finite Group DatabasebetaFinite Group Databasebeta| order | 1250 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C52 · Schur multiplier (H2) = C1 |
| direct product | D5×5−1+2 |
| unambiguous semidirect product | C5⋊(C2×5−1+2) |
| unambiguous semidirect product | (D5×C25)⋊C5 |
| Fitting, Sylow | C5×5−1+2 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C5 | C52×D5 |
| commutator, socle, cosocle | C52 | C2×C52 |
| Fitting | C5×5−1+2 | C2 |
| prime radical | C52×D5 | C5 |
| hypercenter | 5−1+2 | D5 |
| hypocenter | C5 | C2×5−1+2 |
| LCS | G ▷ C52 ▷ C5 | |
| UCS | C1 ◁ C5 ◁ 5−1+2 | |
70 conjugacy classes of subgroups.
| transitive degree | μt(G) = 50 |
116 conjugacy classes — view the full character table.
| maximal subgroup of | D5×D25⋊C5, AGL(1,𝔽5)×5−1+2, #72228845 |
| maximal normal subgroup of | D5×D25⋊C5, AGL(1,𝔽5)×5−1+2, #72228845 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1250, 47);