Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C22×#64241 (order 1152) · Inn(G) = D32 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×C4×D32 |
| Fitting | C33×C4 ⋊ C22 |
| derived | C32 ⋊ C22×C3×C4 |
| Sylow | C33 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | D32 |
| commutator, hypocenter | C32 | C22×C3×C4 |
| Frattini | C2 | C2×C3×D32 |
| Fitting | C33×C4 | C22 |
| socle | C2×C33 | C23 |
| prime radical | C2×C3×D32 | C2 |
| cosocle | C2×C32 | C23×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C3×C4 | |
270 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 48 |
A minimal faithful permutation representation — 3 generators on 13 points:
(5,6,7)(8,9,10)(12,13);(9,10)(11,12,13);(1,4,3,2)108 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D32⋊C8, D32⋊(C3⋊C8), D32×C3⋊C8, … |
| maximal normal subgroup of | C3×D32⋊C8, D32⋊(C3⋊C8), D32×C3⋊C8, … |
| maximal quotient of | C3×D32×C8, C3×D3×ΓL(1,𝔽25), C3×#63791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 648);