Finite Group DatabasebetaFinite Group Databasebeta| order | 340 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C2×C16×AGL(1,𝔽5) (order 640) · Inn(G) = D5 · Out(G) = C22×C16 |
| homology | Abelianization (H1) = C22×C17 · Schur multiplier (H2) = C2 |
| direct product | C2×D5×C17 |
| unambiguous semidirect product | C5⋊(C22×C17) |
| unambiguous semidirect product | (C2×C5)⋊(C2×C17) |
| Fitting, socle | C2×C5×C17 ⋊ C2 |
| derived, Sylow | C5 ⋊ C22×C17 |
| Hall | C5×C17 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C17 | D5 |
| commutator, cosocle, hypocenter | C5 | C22×C17 |
| Frattini | C1 | G |
| Fitting, socle | C2×C5×C17 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C17 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 170 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(4,5)(6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24);(1,2)(3,4,6,7,5)136 conjugacy classes — view the full character table.
| maximal subgroup of | D5×Dic17, #10621754, #10621756, … |
| maximal normal subgroup of | D5×Dic17, #10621754, #10621756, … |
| maximal quotient of | C17×Dic10, C4×D5×C17, C17×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(340, 13);