Finite Group DatabasebetaFinite Group Databasebeta| order | 351 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = C3×C4×#272 (order 648) · Inn(G) = C32 · Out(G) = C3×C4×D3 |
| homology | Abelianization (H1) = C32×C13 · Schur multiplier (H2) = C1 |
| direct product | C13×3−1+2 |
| unambiguous semidirect product | C9⋊(C3×C13) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C13 | C32 |
| commutator, Frattini, cosocle | C3 | C32×C13 |
| prime radical | C32×C13 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C13 ◁ G | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 117 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,4,3,6,8,7,5,9)(10,11,12,13,14,15,16,17,18,19,20,21,22);(2,5,6)(4,8,9)143 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(351, 11);