Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of #677 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×AGL(1,ℤ/8) (order 1280) · Inn(G) = D20 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D40 |
| Fitting | C3×C5×C8 ⋊ C2 |
| Hall | C3×C5 ⋊ D8 |
| Sylow | C5 ⋊ C3×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | D20 |
| commutator, cosocle | C4×C5 | C22×C3 |
| Frattini | C4 | C2×C3×D5 |
| Fitting | C3×C5×C8 | C2 |
| socle | C2×C3×C5 | D4 |
| hypercenter | C3×C8 | D5 |
| hypocenter | C5 | C3×D8 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 ◁ C3×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:
(1,2,3)(4,5,6,7,8,9,10,11)(12,13,14,15,16);(5,11)(6,10)(7,9)(13,16)(14,15)69 conjugacy classes — view the full character table.
| maximal subgroup of | #90937, #90938, #90940, … |
| maximal normal subgroup of | #90937, #90938, #90940, … |
| maximal quotient of | C3×D80, C3×#3932, C3×Dic40, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 36);