Finite Group DatabasebetaFinite Group Databasebeta| order | 1250 |
| isoclinism | isoclinic class of D25⋊C5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C52 · Schur multiplier (H2) = C5 |
| direct product | C5×D25⋊C5 |
| unambiguous semidirect product | D25⋊C52 |
| unambiguous semidirect product | (C5×D25)⋊C5 |
| Fitting, Sylow | C5×5−1+2 ⋊ C2 |
| derived | C25 ⋊ C2×C52 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D25⋊C5 |
| commutator, cosocle, hypocenter | C25 | C2×C52 |
| Frattini | C5 | C52×D5 |
| Fitting | C5×5−1+2 | C2 |
| socle | C52 | C5×D5 |
| LCS | G ▷ C25 | |
| UCS | C1 ◁ C5 | |
76 conjugacy classes of subgroups.
| transitive degree | μt(G) = 50 |
110 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C5×D25⋊C5, C5×AGL(1,ℤ/25), D5×D25⋊C5, … |
| maximal normal subgroup of | C2×C5×D25⋊C5, C5×AGL(1,ℤ/25), D5×D25⋊C5, … |
| maximal quotient of | C2×C5×D25⋊C5, C5×Dic25⋊C5 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1250, 48);