Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of AGL(1,ℤ/9) |
| automorphism | Aut(G) = #120×AGL(1,ℤ/9) (order 1728) · Inn(G) = AGL(1,ℤ/9) · Out(G) = #120 |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C4 |
| direct product | C4×3−1+2⋊C4 |
| unambiguous semidirect product | 3−1+2⋊C42 |
| unambiguous semidirect product | (C4×Dic9)⋊C3 |
| derived | C9 ⋊ C3×C42 |
| Sylow | 3−1+2 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | AGL(1,ℤ/9) |
| commutator, hypocenter | C9 | C3×C42 |
| Frattini, socle | C22×C3 | C2×C3×D3 |
| Fitting | C2×C4×3−1+2 | C2 |
| prime radical | C22×C32 | C2×D3 |
| cosocle | C22×C9 | C22×C3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C4 | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,6,9,4,5,8)(3,7)(14,15,16,17);(1,9,4,7,5,6,3,2,8)(10,11,12,13)80 conjugacy classes — view the full character table.
| maximal subgroup of | C8×3−1+2⋊C4, (3−1+2⋊C4)⋊C8, #63375⋊C3, … |
| maximal normal subgroup of | C8×3−1+2⋊C4, (3−1+2⋊C4)⋊C8, #63375⋊C3, … |
| maximal quotient of | C4×3−1+2⋊C8, #63362⋊C3, C8×3−1+2⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 144);