Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C24×D32 (order 576) · Inn(G) = D32 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×D3×C3⋊C8 |
| unambiguous semidirect product | (C3×D3)⋊(C3×C8) |
| derived | C32 ⋊ C2×C3×C8 |
| Sylow | C33 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | D32 |
| commutator, hypocenter | C32 | C2×C3×C8 |
| Frattini | C4 | C3×D32 |
| Fitting | C33×C4 | C22 |
| socle | C2×C33 | C2×C4 |
| prime radical | C2×C32×D3 | C4 |
| cosocle | C32×C4 | C22×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C3×C4 | |
126 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 52 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 — 2 generators on 17 points:
(1,5,7,2,4,8,3,6)(9,10,11)(12,13,14)(16,17);(13,14)(15,16,17)108 conjugacy classes — view the full character table.
| maximal subgroup of | D32×C3⋊C8, D3×#3710, D3×#3711, … |
| maximal normal subgroup of | D32×C3⋊C8, D3×#3710, D3×#3711, … |
| maximal quotient of | C3×D3×C3⋊C16, C3×#63543, C3×Dic3×C3⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 414);