Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×#120 (order 1280) · Inn(G) = D5 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C4 |
| direct product | C3×C4×Dic5 |
| derived, Sylow | C5 ⋊ C3×C42 |
| Hall | C3×C5 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | D5 |
| commutator, hypocenter | C5 | C3×C42 |
| Frattini | C22 | C2×C3×D5 |
| Fitting | C2×C3×C4×C5 | C2 |
| socle, prime radical | C22×C3×C5 | C22 |
| cosocle | C22×C5 | C22×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C3×C4 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3)(4,5)(10,13,12,11)(14,15,16);(1,2,4,5,3)(6,9,8,7)96 conjugacy classes — view the full character table.
| maximal subgroup of | Dic5×C3⋊C8, #90952, #90979, … |
| maximal normal subgroup of | Dic5×C3⋊C8, #90952, #90979, … |
| maximal quotient of | C3×C4×#168, C3×#3935, C3×C8×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 40);