Finite Group DatabasebetaFinite Group Databasebeta| order | 750 |
| isoclinism | isoclinic class of C3⋊(C5⋊D5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C5 |
| direct product | C5×C3⋊(C5⋊D5) |
| unambiguous semidirect product | C3⋊(C5×C5⋊D5) |
| unambiguous semidirect product | C5⋊(C5×D15) |
| Fitting, socle, Hall | C3×C53 ⋊ C2 |
| derived | C3×C52 ⋊ C2×C5 |
| Sylow | C53 ⋊ D3 |
| Sylow | C3 ⋊ C5×C5⋊D5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | C3⋊(C5⋊D5) |
| commutator, cosocle, hypocenter | C3×C52 | C2×C5 |
| Frattini | C1 | G |
| Fitting, socle | C3×C53 | C2 |
| LCS | G ▷ C3×C52 | |
| UCS | C1 ◁ C5 | |
112 conjugacy classes of subgroups.
| transitive degree | μt(G) = 150 |
195 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D3×C5⋊D5, C5×D5×D15, D5×C3⋊(C5⋊D5), … |
| maximal normal subgroup of | C5×D3×C5⋊D5, C5×D5×D15, D5×C3⋊(C5⋊D5), … |
| maximal quotient of | C5×C3⋊#1039, C2×C5×C3⋊(C5⋊D5) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(750, 37);