Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D3×ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
| direct product | Dic3×ΓL(1,𝔽16) |
| derived, Hall | C32×C5 ⋊ C42 |
| Sylow | C32 ⋊ C4×AGL(1,𝔽5) |
| Sylow | C5 ⋊ Dic32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×ΓL(1,𝔽16) |
| commutator, hypocenter | C32×C5 | C42 |
| Fitting, socle | C2×C32×C5 | C2×C4 |
| prime radical, cosocle | C2×C32×D5 | C22 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
136 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 60 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic3×AGL(1,𝔽5), AGL(1,𝔽5)×#624, #13088408, … |
| maximal normal subgroup of | D3×Dic3×AGL(1,𝔽5), AGL(1,𝔽5)×#624, #13088408, … |
| maximal quotient of | C3⋊C8×ΓL(1,𝔽16), #13085443, #13085444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 439);