Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C23 |
| direct product | D4×C24⋊C5 |
| unambiguous semidirect product | C4⋊(C2×C24⋊C5) |
| unambiguous semidirect product | C22⋊(C2×C24⋊C5) |
| unambiguous semidirect product | (C24×D4)⋊C5 |
| Fitting, Sylow | C24×D4 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C24⋊C5 |
| commutator, socle, cosocle | C25 | C22×C5 |
| Fitting | C24×D4 | C5 |
| hypercenter | D4 | C24⋊C5 |
| hypocenter | C24 | C5×D4 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 ◁ D4 | |
1084 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 20 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | D8×C24⋊C5, SD16×C24⋊C5, C2×D4×C24⋊C5, … |
| maximal normal subgroup of | D8×C24⋊C5, SD16×C24⋊C5, C2×D4×C24⋊C5, … |
| maximal quotient of | C22⋊C4×C24⋊C5, C4⋊C4×C24⋊C5, D4×2−1+4⋊C5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21469);