Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D15 |
| automorphism | Aut(G) = C23×D3×AGL(1,𝔽5) (order 960) · Inn(G) = D15 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C8×D15 |
| Fitting | C3×C5×C8 ⋊ C2 |
| derived, Hall | C3×C5 ⋊ C2×C8 |
| Sylow | C5 ⋊ D3×C8 |
| Sylow | C3 ⋊ C8×D5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | D15 |
| commutator, hypocenter | C3×C5 | C2×C8 |
| Frattini | C4 | C2×D15 |
| Fitting | C3×C5×C8 | C2 |
| socle | C2×C3×C5 | C2×C4 |
| prime radical | C2×D15 | C4 |
| cosocle | C3×C4×C5 | C22 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C8 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3)(4,5)(7,8);(1,2,4,5,3)(6,7,8)(9,10,11,13,12,14,15,16)72 conjugacy classes — view the full character table.
| maximal subgroup of | #90933, #90936, C16×D15, … |
| maximal normal subgroup of | #90933, #90936, C16×D15, … |
| maximal quotient of | C16×D15, #91082, C8×Dic15, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 65);