Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | Aut(G) = C22×D4×AGL(1,ℤ/9) (order 1728) · Inn(G) = C22×D9 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#3699 |
| Fitting | C3×D4×C9 ⋊ C2 |
| Sylow | C3×C9 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×D9 |
| commutator, cosocle | C2×C9 | C23×C3 |
| Frattini | C2×C3 | C22×C3×D3 |
| Fitting | C3×D4×C9 | C2 |
| socle | C2×C32 | C22×D3 |
| hypercenter | C3×D4 | D9 |
| hypocenter | C9 | C3×#41 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2×C3 ◁ C3×D4 | |
136 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3)(4,9,11,10,5,7,6,12,8)(14,18)(15,19);(13,19,16,15)(14,20,18,17);(5,7)(6,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)90 conjugacy classes — view the full character table.
| maximal subgroup of | #11721173, #11721177, #11721181, … |
| maximal normal subgroup of | #11721173, #11721177, #11721181, … |
| maximal quotient of | C3×#63439, C3×#63440, C3×#63441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 357);