Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C3⋊(C5⋊S4) |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C3⋊(C5⋊S4) |
| unambiguous semidirect product | (C2×C3)⋊(C5⋊S4) |
| unambiguous semidirect product | (C2×C5)⋊(C3⋊S4) |
| unambiguous semidirect product | (C2×A4)⋊D15 |
| Fitting, socle | C23×C3×C5 ⋊ D3 |
| derived | C3×C5×A4 ⋊ C22 |
| Sylow | C5 ⋊ C2×C3⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(C5⋊S4) |
| commutator, cosocle, hypocenter | C3×C5×A4 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3×C5 | D3 |
| LCS | G ▷ C3×C5×A4 | |
| UCS | C1 ◁ C2 | |
172 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 90 |
66 conjugacy classes — view the full character table.
| maximal subgroup of | #13088541, #13088542, #13088545, … |
| maximal normal subgroup of | #13088541, #13088542, #13088545, … |
| maximal quotient of | #13089078, C4×C3⋊(C5⋊S4), C4⋊(C3⋊(C5⋊S4)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 817);