Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C22×C3×ΓL(1,𝔽16) |
| unambiguous semidirect product | D5⋊(C2×C3×Dic3) |
| Fitting, socle | C22×C32×C5 ⋊ C4 |
| derived | C3×C5 ⋊ C22×C3×C4 |
| Hall | C32×C5 ⋊ C22×C4 |
| Sylow | C32 ⋊ C22×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C22×C3×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5 | C22×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C5 | C4 |
| prime radical | C22×C32×D5 | C2 |
| cosocle | C3×D5 | C23×C3 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C22×C3 | |
248 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 823);