Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| automorphism | Aut(G) = C2×D3×D4×AGL(1,𝔽5) (order 1920) · Inn(G) = C2×D3×D5 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| direct product | D5×Dic6 |
| unambiguous semidirect product | (C3×D5)⋊Q8 |
| Hall | C3×C5 ⋊ C2×Q8 |
| Sylow | C5 ⋊ C2×Dic6 |
| Sylow | C3 ⋊ Q8×D5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×D5 |
| commutator, socle, cosocle | C2×C3×C5 | C23 |
| Fitting | C3×C4×C5 | C22 |
| prime radical | C2×C3×D5 | C22 |
| hypercenter | C4 | D3×D5 |
| hypocenter | C3×C5 | C2×Q8 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3)(4,6,7,11)(5,8,9,10)(12,13,15,16,14);(2,3)(4,5,7,9)(6,10,11,8);(13,14)(15,16)36 conjugacy classes — view the full character table.
| maximal subgroup of | #91153, #91154, D5×#204, … |
| maximal normal subgroup of | #91153, #91154, D5×#204, … |
| maximal quotient of | #91323, #91328, #91329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 125);