Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| isoclinism | isoclinic class of C24⋊C5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C2×C4×C24⋊C5 |
| unambiguous semidirect product | (C25×C4)⋊C5 |
| Fitting, Sylow | C25×C4 ⋊ C5 |
| derived | C24 ⋊ C2×C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C24⋊C5 |
| commutator, hypocenter | C24 | C2×C4×C5 |
| Frattini | C2 | C22×C24⋊C5 |
| Fitting | C25×C4 | C5 |
| socle | C26 | C2×C5 |
| prime radical | C22×C24⋊C5 | C2 |
| cosocle | C25 | C22×C5 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C2×C4 | |
1084 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 40 |
64 conjugacy classes — view the full character table.
| maximal subgroup of | C42×C24⋊C5, C22⋊C4×C24⋊C5, C4⋊C4×C24⋊C5, … |
| maximal normal subgroup of | C42×C24⋊C5, C22⋊C4×C24⋊C5, C4⋊C4×C24⋊C5, … |
| maximal quotient of | C42×C24⋊C5, C22⋊C4×C24⋊C5, C4⋊C4×C24⋊C5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21468);