Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C33 |
| direct product | C2×C9⋊H(3,𝔽3) |
| unambiguous semidirect product | (C2×C9)⋊H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C33 |
| commutator, Frattini, cosocle | C32 | C2×C33 |
| prime radical | C2×C3×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ G | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(4,6,9)(7,11,10)(13,15,18)(16,20,19);(12,13,16,14,15,19,17,18,20);(3,4,7)(5,6,10)(8,9,11)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 198);