Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D32⋊D3 |
| automorphism | Aut(G) = C22×D3×GO+(4,𝔽2) (order 1728) · Inn(G) = D32⋊D3 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×D32⋊D3 |
| Fitting, socle | C2×C33 ⋊ D4 |
| Sylow | C33 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D32⋊D3 |
| commutator, cosocle | C3×C3⋊D3 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C2×C33 | D4 |
| hypocenter | C33 | C2×D4 |
| LCS | G ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 | |
192 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 11 points:
(3,4,5)(6,11,9,10,8,7);(4,5)(7,10);(1,2)36 conjugacy classes — view the full character table.
| maximal subgroup of | (D32⋊D3)⋊C4, #11724112, #11724113, … |
| maximal normal subgroup of | (D32⋊D3)⋊C4, #11724112, #11724113, … |
| maximal quotient of | #11724298, #11724299, #11724300, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 755);