Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of ΓL(1,𝔽16)⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×ΓL(1,𝔽16)⋊D3 |
| unambiguous semidirect product | D5⋊#624 |
| Fitting, socle | C2×C32×C5 ⋊ C2×C4 |
| derived, Hall | C32×C5 ⋊ C22×C4 |
| Sylow | C32 ⋊ C22×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C2×#624 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽16)⋊D3 |
| commutator, hypocenter | C32×C5 | C22×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | C2×C4 |
| prime radical | C2×D5×C3⋊D3 | C2 |
| cosocle | C32×D5 | C23 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 60 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | #13088347, #13088362, AGL(1,𝔽5)×#624, … |
| maximal normal subgroup of | #13088347, #13088362, AGL(1,𝔽5)×#624, … |
| maximal quotient of | #13088722, #13088723, #13088724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 805);