Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | 5−1+2⋊Q8 |
| unambiguous semidirect product | Dic50⋊C5 |
| Sylow | 5−1+2 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D25⋊C5 |
| commutator, cosocle | C2×C25 | C22×C5 |
| Frattini, socle | C2×C5 | C2×C5×D5 |
| Fitting | C4×5−1+2 | C2 |
| prime radical | C2×C52 | C2×D5 |
| hypercenter | C4 | D25⋊C5 |
| hypocenter | C25 | C5×Q8 |
| LCS | G ▷ C2×C25 ▷ C25 | |
| UCS | C1 ◁ C2 ◁ C4 | |
42 conjugacy classes of subgroups.
| transitive degree | μt(G) = 200 |
Polycyclic presentation:
2^2,5,2,5^2|a^2=d,b^2=d,e^5=f,b^a=bd,e^a=e^4f^4,f^a=f^4,e^c=ef^473 conjugacy classes — view the full character table.
| maximal subgroup of | Dic100⋊C5, #87713⋊C5, #87721⋊C5, … |
| maximal normal subgroup of | Dic100⋊C5, #87713⋊C5, #87721⋊C5, … |
| maximal quotient of | #87718⋊C5, C4⋊(Dic25⋊C5), C2×5−1+2⋊Q8 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 56);