Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of Dic30 |
| automorphism | Aut(G) = D3×AGL(1,𝔽5)×B(4,𝔽2) (order 7680) · Inn(G) = C2×D15 · Out(G) = C4×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×D60 |
| Fitting | C2×C3×C4×C5 ⋊ C2 |
| Hall | C3×C5 ⋊ C2×D4 |
| Sylow | C5 ⋊ C2×D12 |
| Sylow | C3 ⋊ C2×D20 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D15 |
| commutator, cosocle | C2×C3×C5 | C23 |
| Frattini | C2 | C22×D15 |
| Fitting | C2×C3×C4×C5 | C2 |
| socle | C22×C3×C5 | C22 |
| hypercenter | C2×C4 | D15 |
| hypocenter | C3×C5 | C2×D4 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
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)(6,7,8,9)(10,11,13,14,12);(1,2)(4,5)(7,9)(11,12)(13,14);(1,2)66 conjugacy classes — view the full character table.
| maximal subgroup of | #90960, #90968, #90969, … |
| maximal normal subgroup of | #90960, #90968, #90969, … |
| maximal quotient of | #91758, C4×D60, #91763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 177);