Finite Group DatabasebetaFinite Group Databasebeta| order | 340 |
| isoclinism | isoclinic class of D17 |
| automorphism | Aut(G) = C2×C4×AGL(1,𝔽17) (order 2176) · Inn(G) = D17 · Out(G) = C2×C4×C8 |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| direct product | C2×C5×D17 |
| unambiguous semidirect product | C17⋊(C22×C5) |
| unambiguous semidirect product | (C2×C17)⋊(C2×C5) |
| Fitting, socle | C2×C5×C17 ⋊ C2 |
| derived, Sylow | C17 ⋊ C22×C5 |
| Hall | C5×C17 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D17 |
| commutator, cosocle, hypocenter | C17 | C22×C5 |
| Frattini | C1 | G |
| Fitting, socle | C2×C5×C17 | C2 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C2×C5 | |
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)(3,4,5,6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24);(1,2)(8,9,11,13,15,17,19,21,23,24,22,20,18,16,14,12,10)100 conjugacy classes — view the full character table.
| maximal subgroup of | Dic5×D17, #10621754, #10621755, … |
| maximal normal subgroup of | Dic5×D17, #10621754, #10621755, … |
| maximal quotient of | C5×Dic34, C4×C5×D17, C5×D68, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(340, 12);