Finite Group DatabasebetaFinite Group Databasebeta| order | 340 |
| isoclinism | isoclinic class of #66774 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C5×C17 ⋊ C4 |
| Sylow | C17 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ Dic17 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C17 | C4 |
| prime radical, cosocle | D5×C17 | C2 |
| LCS | G ▷ C5×C17 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 85 |
A minimal faithful permutation representation — 2 generators on 22 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(19,20,21,22);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,19,22,20,21)Polycyclic presentation:
2^2,5,17|a^2=b,c^a=c^3,d^a=d^16,c^b=c^437 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×D17, C2×#66774, C3×#66774, … |
| maximal normal subgroup of | AGL(1,𝔽5)×D17, C2×#66774, C3×#66774, … |
| maximal quotient of | #10621748, C2×#66774, C3×#66774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(340, 8);