Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | Aut(G) = C2×D4×((C3⋊D3)⋊C3)⋊C2 (order 1728) · Inn(G) = C22×(C3⋊D3)⋊C3 · Out(G) = C23 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | D4×(C3⋊D3)⋊C3 |
| unambiguous semidirect product | (D4×C3⋊D3)⋊C3 |
| Fitting | D4×H(3,𝔽3) ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×(C3⋊D3)⋊C3 |
| commutator, cosocle | C2×C32 | C23×C3 |
| Frattini, socle | C2×C3 | C22×C3×D3 |
| Fitting | D4×H(3,𝔽3) | C2 |
| hypercenter | D4 | (C3⋊D3)⋊C3 |
| hypocenter | C32 | C2×C3×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ D4 | |
205 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 205 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 36 |
A minimal faithful permutation representation — 3 generators on 13 points:
(6,11,13,8,9,12)(7,10);(2,4)(5,6,8)(7,9,12)(10,13,11);(1,2)(3,4)50 conjugacy classes — view the full character table.
| maximal subgroup of | #11721194, #11721196, #11721198, … |
| maximal normal subgroup of | #11721194, #11721196, #11721198, … |
| maximal quotient of | #64087⋊C3, C22⋊C4×(C3⋊D3)⋊C3, #64090⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 360);