Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽5) (order 640) · Inn(G) = C22×D5 · Out(G) = C24 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×D4×D5 |
| Fitting | C3×C5×D4 ⋊ C2 |
| Hall | C3×C5 ⋊ C2×D4 |
| Sylow | C5 ⋊ C2×C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×D5 |
| commutator, cosocle | C2×C5 | C23×C3 |
| Frattini | C2 | C22×C3×D5 |
| Fitting | C3×C5×D4 | C2 |
| socle | C2×C3×C5 | C23 |
| hypercenter | C3×D4 | D5 |
| hypocenter | C5 | C2×C3×D4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C3 ◁ C3×D4 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,3)(4,5)(10,11,12);(1,2,4,5,3)(7,9);(6,7)(8,9)60 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#4007, #91236, D5×#213, … |
| maximal normal subgroup of | C3×#4007, #91236, D5×#213, … |
| maximal quotient of | C3×#4024, C3×D5×C22⋊C4, C3×#4027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 159);