Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) order 53760 |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C23×C3×D5 |
| unambiguous semidirect product | C5⋊(C24×C3) |
| unambiguous semidirect product | (C2×C5)⋊(C23×C3) |
| Fitting, socle | C23×C3×C5 ⋊ C2 |
| derived, Sylow | C5 ⋊ C24×C3 |
| Hall | C3×C5 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | D5 |
| commutator, cosocle, hypocenter | C5 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3×C5 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C23×C3 | |
268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2)(3,4)(5,6)(7,8,9)(11,12)(13,14);(5,6)(10,11,13,14,12);(3,4);(1,2)96 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6665, D5×#217, #91563, … |
| maximal normal subgroup of | C2×#6665, D5×#217, #91563, … |
| maximal quotient of | C22×C3×Dic10, C22×C3×C4×D5, C22×C3×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 205);