Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of #686 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽5) (order 640) · Inn(G) = C22⋊D5 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#686 |
| Fitting | C3×C5×D4 ⋊ C2 |
| Hall | C3×C5 ⋊ D8 |
| Sylow | C5 ⋊ C3×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22⋊D5 |
| commutator, cosocle | C4×C5 | C22×C3 |
| Frattini | C4 | C2×C3×D5 |
| Fitting | C3×C5×D4 | C2 |
| socle | C2×C3×C5 | D4 |
| hypercenter | C3×D4 | D5 |
| hypocenter | C5 | C3×D8 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,5,3)(7,10)(8,13)(11,12)(14,15,16);(2,3)(4,5)(6,7)(8,10)(9,12)(11,13)51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#686, #91480, #91481, … |
| maximal normal subgroup of | D3×#686, #91480, #91481, … |
| maximal quotient of | C3×#3939, C3×#3941, C3×#3958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 44);