Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | Aut(G) = C2×D4×AGL(1,ℤ/9) (order 864) · Inn(G) = C22×AGL(1,ℤ/9) · Out(G) = C22 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | D4×AGL(1,ℤ/9) |
| unambiguous semidirect product | D9⋊(C3×D4) |
| unambiguous semidirect product | (D4×D9)⋊C3 |
| Fitting | D4×3−1+2 ⋊ C2 |
| Sylow | 3−1+2 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×AGL(1,ℤ/9) |
| commutator, cosocle | C2×C9 | C23×C3 |
| Frattini, socle | C2×C3 | C22×C3×D3 |
| Fitting | D4×3−1+2 | C2 |
| hypercenter | D4 | AGL(1,ℤ/9) |
| hypocenter | C9 | C2×C3×D4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ D4 | |
170 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 13 points:
(2,6,9,4,5,8)(3,7);(1,5,8,7,2,4,3,9,6)(11,13);(10,11)(12,13)50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 362);