Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of ΓL(1,𝔽16) |
| automorphism | Aut(G) = D3×D4×AGL(1,𝔽5) (order 960) · Inn(G) = ΓL(1,𝔽16) · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C3⋊#170 |
| unambiguous semidirect product | (C2×C3)⋊#170 |
| derived, Hall | C3×C5 ⋊ C2×C8 |
| Sylow | C5 ⋊ C2×C3⋊C8 |
| Sylow | C3 ⋊ C2×#170 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5 | C2×C8 |
| Frattini | C2 | C2×ΓL(1,𝔽16) |
| Fitting, socle, prime radical | C22×C3×C5 | C4 |
| cosocle | C3×Dic5 | C22 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C22 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 18 points:
(4,5)(7,8,9,10)(11,12,13,15,14,16,17,18);(1,2)(3,4,5)(6,7,10,8,9)36 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#170, #91169, (C2×D3)⋊#170, … |
| maximal normal subgroup of | Dic3×#170, #91169, (C2×D3)⋊#170, … |
| maximal quotient of | C2×C3⋊#674, #91227, C4×C3⋊#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 122);