Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D3×D5 |
| automorphism | Aut(G) = D3×AGL(1,𝔽5)×#120 (order 3840) · Inn(G) = D3×D5 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#1200 |
| derived, Hall | C3×C5 ⋊ C22×C4 |
| Sylow | C5 ⋊ C2×C4×D3 |
| Sylow | C3 ⋊ C2×C4×D5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×D5 |
| commutator, hypocenter | C3×C5 | C22×C4 |
| Frattini | C2 | C2×D3×D5 |
| Fitting, socle | C22×C3×C5 | C22 |
| prime radical | C22×D15 | C2 |
| cosocle | C2×C3×C5 | C23 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C22 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(4,5)(6,8,9,7)(10,11,13,14,12);(3,4,5)(6,7,9,8)(11,12)(13,14);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | #91171, #91356, #91395, … |
| maximal normal subgroup of | #91171, #91356, #91395, … |
| maximal quotient of | C2×#6648, #91290, #91291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 144);