Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C22×D4×AGL(1,ℤ/9) (order 1728) · Inn(G) = D9 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×C9⋊C8 |
| derived | C9 ⋊ C2×C3×C8 |
| Sylow | C3×C9 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | D9 |
| commutator, hypocenter | C9 | C2×C3×C8 |
| Frattini | C3×C4 | C2×C3×D3 |
| Fitting | C2×C3×C4×C9 | C2 |
| socle, prime radical | C22×C32 | Dic3 |
| cosocle | C4×C9 | C22×C3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C3×C4 | |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2)(3,4,5)(6,7,8,10,9,11,12,13)(15,17)(16,20)(18,19)(21,22);(1,2)(14,22,17,20,19,18,16,15,21)144 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×C9⋊C8, #11720258, Dic3⋊(C9⋊C8), … |
| maximal normal subgroup of | Dic3×C9⋊C8, #11720258, Dic3⋊(C9⋊C8), … |
| maximal quotient of | C3×C4×C9⋊C8, C3×C4⋊(C9⋊C8), C2×C3×C9⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 124);