Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D3×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×D3×S4 |
| unambiguous semidirect product | (C3×S4)⋊(C2×C5) |
| unambiguous semidirect product | (C3⋊S4)⋊(C2×C5) |
| unambiguous semidirect product | (D3×A4)⋊(C2×C5) |
| Fitting, socle | C22×C3×C5 ⋊ C2×D3 |
| derived | C3×A4 ⋊ C22×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D3×S4 |
| commutator, cosocle, hypocenter | C3×A4 | C22×C5 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C5 | C2×D3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C5 | |
140 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
75 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D5×S4, C2×C5×D3×S4, C5×D3×#1020, … |
| maximal normal subgroup of | D3×D5×S4, C2×C5×D3×S4, C3×C5×D3×S4 |
| maximal quotient of | C5×#64196, C5×#64197, C5×#64198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 792);