Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D3×AGL(1,𝔽5) |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽5) (order 480) · Inn(G) = D3×AGL(1,𝔽5) · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| Fitting, socle | C2×C3×C5 ⋊ C2×C4 |
| derived, Hall | C3×C5 ⋊ C22×C4 |
| Sylow | C5 ⋊ C2×C4×D3 |
| Sylow | C3 ⋊ C22×AGL(1,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×AGL(1,𝔽5) |
| commutator, hypocenter | C3×C5 | C22×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5 | C2×C4 |
| prime radical | C2×D3×D5 | C2 |
| cosocle | C3×D5 | C23 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 3 generators on 10 points:
(3,4,5)(7,8,9,10);(1,2)(4,5)(6,7,8,10,9);(1,2)30 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 195);