Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of A42 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C5 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C5×A42 |
| unambiguous semidirect product | (C23×A4)⋊(C3×C5) |
| unambiguous semidirect product | (C23×C5×A4)⋊C3 |
| unambiguous semidirect product | (C2×C5×C22⋊A4)⋊C3 |
| Fitting, socle, Hall | C25×C5 ⋊ C32 |
| derived | C24 ⋊ C2×C32×C5 |
| Sylow | C25 ⋊ C32×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | A42 |
| commutator, cosocle, hypocenter | C24 | C2×C32×C5 |
| Frattini | C1 | G |
| Fitting, socle | C25×C5 | C32 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C2×C5 | |
244 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 90 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D5×A42, C2×C5×A4⋊S4, C2×C5⋊(A4⋊S4), … |
| maximal normal subgroup of | C2×D5×A42, C2×C5×A4⋊S4, C2×C5⋊(A4⋊S4), … |
| maximal quotient of | A42×AGL(1,𝔽11), C5×D5×A42, C22×C5×A42, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5909);