Finite Group DatabasebetaFinite Group Databasebeta| order | 405 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C5 · Schur multiplier (H2) = C34 |
| direct product | C3×C5×H(3,𝔽3) |
| unambiguous semidirect product | C32⋊(C32×C5) |
| unambiguous semidirect product | (C32×C5)⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C5 | C32 |
| commutator, Frattini, cosocle | C3 | C33×C5 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C5 ◁ G | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 135 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3)(13,14,15,16,17);(4,5,7)(6,9,11)(8,12,10);(5,8,9)(7,11,12)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(405, 12);