Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽5) (order 640) · Inn(G) = D5 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×#168 |
| derived, Sylow | C5 ⋊ C2×C3×C8 |
| Hall | C3×C5 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | D5 |
| commutator, hypocenter | C5 | C2×C3×C8 |
| Frattini | C4 | C2×C3×D5 |
| Fitting | C2×C3×C4×C5 | C2 |
| socle, prime radical | C22×C3×C5 | C4 |
| cosocle | C4×C5 | C22×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C3×C4 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(3,4,5)(7,8)(9,10)(11,12,13,15,14,16,17,18);(1,2)(6,7,9,10,8)96 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#168, #90951, #90953, … |
| maximal normal subgroup of | Dic3×#168, #90951, #90953, … |
| maximal quotient of | C3×C4×#168, C3×C4⋊#168, C2×C3×#672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 38);