Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C42×D3 (order 192) · Inn(G) = D3 · Out(G) = C2×C42 |
| homology | Abelianization (H1) = C5×C16 · Schur multiplier (H2) = C1 |
| direct product | C5×C3⋊C16 |
| unambiguous semidirect product | C3⋊(C5×C16) |
| derived, Sylow | C3 ⋊ C5×C16 |
| Hall | C3×C5 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C8 | D3 |
| commutator, hypocenter | C3 | C5×C16 |
| Frattini | C8 | C5×D3 |
| Fitting | C3×C5×C8 | C2 |
| socle, prime radical | C2×C3×C5 | C8 |
| cosocle | C3×C8 | C2×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C5×C8 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,3)(4,10,13,19,7,11,18,9,8,17,6,15,14,5,12,16)(20,21,22,23,24);(1,2,3)120 conjugacy classes — view the full character table.
| maximal subgroup of | D5×C3⋊C16, #90933, #90934, … |
| maximal normal subgroup of | D5×C3⋊C16, #90933, #90934, … |
| maximal quotient of | C5×C3⋊C32, C2×C5×C3⋊C16, C5×C9⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 1);