Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C2×D4×AGL(1,ℤ/9) (order 864) · Inn(G) = D9 · Out(G) = C2×C3×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C9⋊C8 |
| unambiguous semidirect product | (C2×C9)⋊C8 |
| derived, Sylow | C9 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D9 |
| commutator, hypocenter | C9 | C2×C8 |
| Frattini | C3×C4 | C2×D3 |
| Fitting | C2×C4×C9 | C2 |
| socle, prime radical | C22×C3 | Dic3 |
| cosocle | C4×C9 | C22 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C4 | |
33 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2)(3,4,5,7,6,8,9,10)(12,14)(13,17)(15,16)(18,19);(1,2)(11,19,14,17,16,15,13,12,18)48 conjugacy classes — view the full character table.
| maximal subgroup of | C4×C9⋊C8, #63362, C4⋊(C9⋊C8), … |
| maximal normal subgroup of | C4×C9⋊C8, #63362, C4⋊(C9⋊C8), … |
| maximal quotient of | C4×C9⋊C8, C4⋊(C9⋊C8), C2×C9⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 9);