Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D32×AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | D32×AGL(1,𝔽5) |
| unambiguous semidirect product | (C3×D3×AGL(1,𝔽5))⋊C2 |
| unambiguous semidirect product | (C3⋊D3×AGL(1,𝔽5))⋊C2 |
| unambiguous semidirect product | (D3×ΓL(1,𝔽16))⋊C2 |
| Fitting, socle, derived, Hall | C32×C5 ⋊ C22×C4 |
| Sylow | C32 ⋊ C22×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C4×D32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C5 | C22×C4 |
| prime radical | D32×D5 | C2 |
| cosocle | C32×D5 | C23 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 | |
232 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 30 |
45 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 783);