Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of #204 |
| automorphism | Aut(G) = C4×D3×AGL(1,ℤ/8) (order 768) · Inn(G) = D12 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×D24 |
| Fitting | C3×C5×C8 ⋊ C2 |
| Hall | C3×C5 ⋊ D8 |
| Sylow | C3 ⋊ C5×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | D12 |
| commutator, cosocle | C3×C4 | C22×C5 |
| Frattini | C4 | C2×C5×D3 |
| Fitting | C3×C5×C8 | C2 |
| socle | C2×C3×C5 | D4 |
| hypercenter | C5×C8 | D3 |
| hypocenter | C3 | C5×D8 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C5 ◁ C4×C5 ◁ C5×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);(7,13)(8,12)(9,11)(15,16)75 conjugacy classes — view the full character table.
| maximal subgroup of | #90937, #90939, #90941, … |
| maximal normal subgroup of | #90937, #90939, #90941, … |
| maximal quotient of | C5×D48, C5×#800, C5×Dic24, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 52);