Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D3×A4 |
| automorphism | Aut(G) = C2×C3×D3×S4 (order 864) · Inn(G) = D3×A4 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C22 |
| direct product | C2×D3×C22⋊C9 |
| unambiguous semidirect product | C3⋊(C22×C22⋊C9) |
| unambiguous semidirect product | (C2×C3)⋊(C2×C22⋊C9) |
| unambiguous semidirect product | (C23×D3)⋊C9 |
| derived | C22×C3 ⋊ C22×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D3×A4 |
| commutator, hypocenter | C22×C3 | C22×C9 |
| Frattini | C3 | C2×D3×A4 |
| Fitting, socle | C23×C32 | C2×C3 |
| prime radical | C23×C3×D3 | C3 |
| cosocle | C22×C32 | C22×C3 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C2×C3 | |
148 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(4,5)(7,8,9)(10,14,18,12,17,13,15,11,16);(1,2)(3,4,5)(6,9)(7,8)72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×(C22⋊C9)⋊C4, #11722266, #11722267, … |
| maximal normal subgroup of | D3×(C22⋊C9)⋊C4, #11722266, #11722267, … |
| maximal quotient of | Dic6×C22⋊C9, C4×D3×C22⋊C9, D12×C22⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 541);