Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D32×D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×D32×D5 |
| unambiguous semidirect product | (C3×D3×D5)⋊C22 |
| unambiguous semidirect product | (D5×C3⋊D3)⋊C22 |
| unambiguous semidirect product | (D3×D15)⋊C22 |
| Fitting, socle | C2×C32×C5 ⋊ C23 |
| derived, Hall | C32×C5 ⋊ C24 |
| Sylow | C32 ⋊ C23×D5 |
| Sylow | C5 ⋊ C22×D32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D32×D5 |
| commutator, cosocle, hypocenter | C32×C5 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | C23 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
568 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 60 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×(C2×D3)⋊AGL(1,𝔽5), #13088405, D5×D32⋊C4, … |
| maximal normal subgroup of | D3×(C2×D3)⋊AGL(1,𝔽5), #13088405, D5×D32⋊C4, … |
| maximal quotient of | D3×D5×Dic6, D5×#3795, D5×#3796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 813);