Finite Group DatabasebetaFinite Group Databasebeta| order | 297 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = C2×C5×#272 (order 540) · Inn(G) = C32 · Out(G) = C2×C5×D3 |
| homology | Abelianization (H1) = C32×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×3−1+2 |
| unambiguous semidirect product | C9⋊(C3×C11) |
| unambiguous semidirect product | (C9×C11)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C11 | C32 |
| commutator, Frattini, cosocle | C3 | C32×C11 |
| prime radical | C32×C11 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C11 ◁ G | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 99 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,4,3,6,8,7,5,9)(10,11,12,13,14,15,16,17,18,19,20);(2,5,6)(4,8,9)121 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(297, 4);