Finite Group DatabasebetaFinite Group Databasebeta| order | 600 |
| isoclinism | isoclinic class of ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C8 · Schur multiplier (H2) = C1 |
| direct product | C5×C3⋊#170 |
| unambiguous semidirect product | C3⋊(C5×#170) |
| derived | C3×C5 ⋊ C5×C8 |
| Hall | C3×C52 ⋊ C8 |
| Sylow | C52 ⋊ C3⋊C8 |
| Sylow | C3 ⋊ C5×#170 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5 | C5×C8 |
| Frattini | C2 | C5×ΓL(1,𝔽16) |
| Fitting, socle, prime radical | C2×C3×C52 | C4 |
| cosocle | C3×Dic5 | C2×C5 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2×C5 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 120 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D3×#170, C5×#6738, C5×#6739, … |
| maximal normal subgroup of | C5×D3×#170, C5×#6738, C5×#6739, … |
| maximal quotient of | C5×C3⋊#674, C2×C5×C3⋊#170, C5×C9⋊#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(600, 76);