Finite Group DatabasebetaFinite Group Databasebeta| order | 1250 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C52 · Schur multiplier (H2) = C52 |
| direct product | D5×H(3,𝔽5) |
| unambiguous semidirect product | C5⋊(C2×H(3,𝔽5)) |
| unambiguous semidirect product | (C52×D5)⋊C5 |
| Fitting, Sylow | C5×H(3,𝔽5) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C5 | C52×D5 |
| commutator, socle, cosocle | C52 | C2×C52 |
| Fitting | C5×H(3,𝔽5) | C2 |
| hypercenter | H(3,𝔽5) | D5 |
| hypocenter | C5 | C2×H(3,𝔽5) |
| LCS | G ▷ C52 ▷ C5 | |
| UCS | C1 ◁ C5 ◁ H(3,𝔽5) | |
110 conjugacy classes of subgroups.
| transitive degree | μt(G) = 50 |
116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1250, 42);