Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D3×GO+(4,𝔽2) |
| automorphism | Aut(G) = D3×AΓL(1,𝔽9) (order 864) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| Fitting, socle, Sylow | C33 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | C3×C3⋊D3 | C23 |
| Fitting, socle, hypocenter | C33 | C2×D4 |
| LCS | G ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 | |
192 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 3 generators on 9 points:
(1,2,3)(4,6)(5,8,9);(4,5)(6,9)(7,8);(2,3)27 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 741);