Finite Group DatabasebetaFinite Group Databasebeta| order | 135 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = C4×#272 (order 216) · Inn(G) = C32 · Out(G) = C4×D3 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽5)×#13091120 |
| homology | Abelianization (H1) = C32×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×3−1+2 |
| unambiguous semidirect product | C9⋊(C3×C5) |
| unambiguous semidirect product | (C5×C9)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C5 | C32 |
| commutator, Frattini, cosocle | C3 | C32×C5 |
| prime radical | C32×C5 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C5 ◁ G | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 45 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2,4,3,6,8,7,5,9)(10,11,12,13,14);(2,5,6)(4,8,9)55 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(135, 4);