Finite Group DatabasebetaFinite Group Databasebeta| order | 600 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C25 · Schur multiplier (H2) = C2 |
| direct product | C2×A4×C25 |
| unambiguous semidirect product | C23⋊(C3×C25) |
| unambiguous semidirect product | (C23×C25)⋊C3 |
| Fitting, Hall | C23×C25 ⋊ C3 |
| derived | C22 ⋊ C2×C3×C25 |
| Sylow | C23 ⋊ C3×C25 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C25 | A4 |
| commutator, hypocenter | C22 | C2×C3×C25 |
| Frattini | C5 | C2×C5×A4 |
| Fitting | C23×C25 | C3 |
| socle | C23×C5 | C3×C5 |
| prime radical | C2×C5×A4 | C5 |
| cosocle | C22×C5 | C2×C3×C5 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C2×C25 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 150 |
200 conjugacy classes — view the full character table.
| maximal subgroup of | C25×SL(2,ℤ/4), A4⋊Dic25, A4×Dic25, … |
| maximal normal subgroup of | C25×SL(2,ℤ/4), A4⋊Dic25, A4×Dic25, … |
| maximal quotient of | C4×A4×C25, C2×SL(2,𝔽3)×C25, C25×#41⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(600, 40);