Finite Group DatabasebetaFinite Group Databasebeta| order | 405 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = C3×C4×#4183 (order 1944) · Inn(G) = C32 · Out(G) = C4×#272 |
| homology | Abelianization (H1) = C3×C5×C9 · Schur multiplier (H2) = C3 |
| direct product | C5×C9⋊C9 |
| unambiguous semidirect product | C9⋊(C5×C9) |
| unambiguous semidirect product | (C5×C9)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C32×C5 | C32 |
| commutator | C3 | C3×C5×C9 |
| Frattini, cosocle | C32 | C32×C5 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C5 ◁ G | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 405 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,4,3,5,7,6,8,9)(11,13,16)(14,18,17)(19,20,21,22,23);(10,11,14,12,13,17,15,16,18)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(405, 4);