Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = C2×D32 · Out(G) = C24 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×D3×Dic6 |
| unambiguous semidirect product | (C3×D3)⋊(C3×Q8) |
| Sylow | C33 ⋊ C2×Q8 |
198 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 198 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) = 17 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,8,4,3)(2,7,6,5)(9,10,11)(12,13,14)(15,16,17);(1,6,4,2)(3,5,8,7)(16,17);(13,14)81 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3715, D3×#3717, D3×#3718, … |
| maximal normal subgroup of | D3×#3715, D3×#3717, D3×#3718, … |
| maximal quotient of | C3×#63837, C3×#63839, C3×#63840, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 642);