Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×#120 (order 1280) · Inn(G) = D5 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×C4×D5 |
| Fitting | C2×C3×C4×C5 ⋊ C2 |
| derived, Sylow | C5 ⋊ C22×C3×C4 |
| Hall | C3×C5 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | D5 |
| commutator, hypocenter | C5 | C22×C3×C4 |
| Frattini | C2 | C22×C3×D5 |
| Fitting | C2×C3×C4×C5 | C2 |
| socle | C22×C3×C5 | C22 |
| prime radical | C22×C3×D5 | C2 |
| cosocle | C2×C5 | C23×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C3×C4 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(3,4,5)(11,12)(13,14);(6,7,8,9)(10,11,13,14,12);(1,2)96 conjugacy classes — view the full character table.
| maximal subgroup of | #90955, C3×#3952, C3×#4003, … |
| maximal normal subgroup of | #90955, C3×#3952, C3×#4003, … |
| maximal quotient of | C3×C4×Dic10, C3×C42×D5, C3×#4018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 156);