Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D3×D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C4×D3×D9 |
| Fitting | C3×C4×C9 ⋊ C22 |
| derived, Sylow | C3×C9 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D3×D9 |
| commutator, hypocenter | C3×C9 | C22×C4 |
| Frattini | C2×C3 | C2×D32 |
| Fitting | C3×C4×C9 | C22 |
| socle | C2×C32 | C22×D3 |
| prime radical | C2×D3×D9 | C2 |
| cosocle | C2×C3×C9 | C23 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C4 | |
178 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3)(5,7)(6,10)(8,11)(9,12);(2,3)(4,9,7,6,11,8,10,5,12);(13,14,15,16)72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C8×D9, D9×ΓL(1,𝔽25), D3×#3663, … |
| maximal normal subgroup of | D3×C8×D9, D9×ΓL(1,𝔽25), D3×#3663, … |
| maximal quotient of | D3×C8×D9, D9×ΓL(1,𝔽25), D3×#3663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 290);