Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C4×D3×B(4,𝔽2) (order 1536) · Inn(G) = C2×D3 · Out(G) = C4×#120 |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C2×C5×D12 |
| Fitting | C2×C3×C4×C5 ⋊ C2 |
| Hall | C3×C5 ⋊ C2×D4 |
| Sylow | C3 ⋊ C2×C5×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5 | C2×D3 |
| commutator, cosocle | C2×C3 | C23×C5 |
| Frattini | C2 | C22×C5×D3 |
| Fitting | C2×C3×C4×C5 | C2 |
| socle | C22×C3×C5 | C22 |
| hypercenter | C2×C4×C5 | D3 |
| hypocenter | C3 | C2×C5×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5 | |
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:
(3,4,5)(6,7,8,9)(10,11,12,13,14);(1,2)(4,5)(7,9);(1,2)90 conjugacy classes — view the full character table.
| maximal subgroup of | #90959, #90966, #90967, … |
| maximal normal subgroup of | #90959, #90966, #90967, … |
| maximal quotient of | C5×#869, C4×C5×D12, C5×#874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 167);