Finite Group DatabasebetaFinite Group Databasebeta| order | 405 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5×C9 · Schur multiplier (H2) = C32 |
| direct product | C5×C32⋊C9 |
| unambiguous semidirect product | C32⋊(C5×C9) |
| unambiguous semidirect product | (C32×C5)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C5 | C32 |
| commutator | C3 | C3×C5×C9 |
| Frattini, cosocle | C32 | C32×C5 |
| prime radical | C33×C5 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C5 ◁ G | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 135 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,4,3,5,7,6,8,9)(10,11,13)(12,15,17)(14,18,16)(19,20,21,22,23);(11,14,15)(13,17,18)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(405, 3);