Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C4×D3×#120 (order 768) · Inn(G) = D3 · Out(G) = C4×#120 |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C4 |
| direct product | C4×C5×Dic3 |
| unambiguous semidirect product | C3⋊(C42×C5) |
| derived, Sylow | C3 ⋊ C42×C5 |
| Hall | C3×C5 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C5 | D3 |
| commutator, hypocenter | C3 | C42×C5 |
| Frattini | C22 | C2×C5×D3 |
| Fitting | C2×C3×C4×C5 | C2 |
| socle, prime radical | C22×C3×C5 | C22 |
| cosocle | C22×C3 | C22×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C4×C5 | |
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)(4,5,6,7,8)(13,14,15,16);(1,2,3)(9,10,11,12)120 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#168, #90953, #90977, … |
| maximal normal subgroup of | Dic3×#168, #90953, #90977, … |
| maximal quotient of | C4×C5×C3⋊C8, C5×#803, C5×C8×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 56);