Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of Dic30 |
| automorphism | Aut(G) = D3×AGL(1,𝔽5)×#120 (order 3840) · Inn(G) = C2×D15 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#1220 |
| Fitting | C23×C3×C5 ⋊ C2 |
| Hall | C3×C5 ⋊ C2×D4 |
| Sylow | C5 ⋊ C2×#67 |
| Sylow | C3 ⋊ C2×C22⋊D5 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D15 |
| commutator, cosocle | C2×C3×C5 | C23 |
| Frattini | C2 | C22×D15 |
| Fitting | C23×C3×C5 | C2 |
| socle | C22×C3×C5 | C22 |
| hypercenter | C23 | D15 |
| hypocenter | C3×C5 | C2×D4 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C22 ◁ C23 | |
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:
(3,4,5)(7,9)(10,11,13,14,12);(4,5)(6,7)(8,9)(11,12)(13,14);(1,2)66 conjugacy classes — view the full character table.
| maximal subgroup of | #90997, #91090, #91526, … |
| maximal normal subgroup of | #90997, #91090, #91526, … |
| maximal quotient of | C2×#6712, #91813, C2×#6714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 184);