Finite Group DatabasebetaFinite Group Databasebeta| order | 750 |
| isoclinism | isoclinic class of C3⋊(D25⋊C5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(D25⋊C5) |
| unambiguous semidirect product | 5−1+2⋊D3 |
| unambiguous semidirect product | D75⋊C5 |
| Fitting, Hall | C3×5−1+2 ⋊ C2 |
| derived | C3×C25 ⋊ C2×C5 |
| Sylow | 5−1+2 ⋊ D3 |
| Sylow | C3 ⋊ D25⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C25 | C2×C5 |
| Frattini | C5 | C5×D15 |
| Fitting | C3×5−1+2 | C2 |
| socle | C3×C5 | C5×D5 |
| LCS | G ▷ C3×C25 | |
| UCS | C1 | |
28 conjugacy classes of subgroups.
| transitive degree | μt(G) = 75 |
Polycyclic presentation:
2,5,3,5^2|d^5=e,c^a=c^2,d^a=d^4e^4,e^a=e^4,d^b=de51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D25⋊C5, C2×C3⋊(D25⋊C5), #72228866, … |
| maximal normal subgroup of | D3×D25⋊C5, C2×C3⋊(D25⋊C5), C3×C3⋊(D25⋊C5) |
| maximal quotient of | C3⋊(Dic25⋊C5), C2×C3⋊(D25⋊C5), #72228866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(750, 20);