Finite Group DatabasebetaFinite Group Databasebeta| order | 1250 |
| isoclinism | isoclinic class of (C5⋊D5)⋊C5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C25 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | (C5⋊D5)⋊C25 |
| Fitting, Sylow | C52⋊C25 ⋊ C2 |
| derived | C52 ⋊ C2×C25 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | (C5⋊D5)⋊C5 |
| commutator, hypocenter | C52 | C2×C25 |
| Frattini, socle | C52 | C5×D5 |
| Fitting | C52⋊C25 | C2 |
| prime radical | C5×C5⋊D5 | C5 |
| cosocle | C53 | C2×C5 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C5 | |
34 conjugacy classes of subgroups.
| transitive degree | μt(G) = 50 |
Polycyclic presentation:
2,5^4|b^5=e,c^a=c^4,d^a=d^4,c^b=cd110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1250, 19);