Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of S4×D15 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | S4×D15 |
| unambiguous semidirect product | (C3×S4)⋊D5 |
| unambiguous semidirect product | (C5×S4)⋊D3 |
| Fitting, socle | C22×C3×C5 ⋊ C2×D3 |
| derived | C3×C5×A4 ⋊ C22 |
| Sylow | C5 ⋊ D3×S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C5×A4 | C22 |
| Fitting, socle | C22×C3×C5 | C2×D3 |
| LCS | G ▷ C3×C5×A4 | |
| UCS | C1 | |
140 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 21 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 |
45 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D5×S4, C2×S4×D15, D15×#1020, … |
| maximal normal subgroup of | D3×D5×S4, C2×S4×D15, C3×S4×D15 |
| maximal quotient of | #13088546, #13088547, #13088548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 794);