Finite Group DatabasebetaFinite Group Databasebeta| order | 750 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #6953⋊C3 |
| Fitting, derived, Sylow | H(3,𝔽5) ⋊ C2×C3 |
| Hall | H(3,𝔽5)⋊C3 ⋊ C2 |
| Hall | #6953 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C5 | (C5⋊D5)⋊C3 |
| commutator, Fitting, cosocle, hypocenter | H(3,𝔽5) | C2×C3 |
| LCS | G ▷ H(3,𝔽5) | |
| UCS | C1 ◁ C5 | |
20 conjugacy classes of subgroups.
| transitive degree | μt(G) = 75 |
Polycyclic presentation:
2,3,5^3|c^a=c^4,d^a=d^4e^2,c^b=c^3d^3e^4,d^b=c^4de^2,d^c=de34 conjugacy classes — view the full character table.
| maximal subgroup of | #13093734, #92537⋊C3, #10625285⋊C2, … |
| maximal normal subgroup of | #13093734, #92537⋊C3, #10625285⋊C2, … |
| maximal quotient of | #92525⋊C3, C2×#6953⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(750, 6);