Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | Aut(G) = C24×D32 (order 576) · Inn(G) = C2×D32 · Out(G) = C23 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×D3×#67 |
| Fitting | C22×C33 ⋊ C22 |
| Sylow | C33 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D32 |
| commutator, cosocle | C2×C32 | C23×C3 |
| Frattini | C2 | C2×C3×D32 |
| Fitting | C22×C33 | C22 |
| socle | C2×C33 | C23 |
| hypercenter | C22×C3 | D32 |
| hypocenter | C32 | C2×C3×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
290 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 13 points:
(2,4)(5,6,7)(8,9,10)(11,12,13);(1,2)(3,4)(12,13);(9,10)81 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3804, D3×#3805, #11724414, … |
| maximal normal subgroup of | D3×#3804, D3×#3805, #11724414, … |
| maximal quotient of | C3×#63839, C3×#63850, C3×#63851, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 658);