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) = C22×C3 |
| unambiguous semidirect product | #3830⋊C3 |
| Fitting | D4×H(3,𝔽3) ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #41 |
| 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 | C3×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ D4 | |
156 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 156 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) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(2,6)(3,7)(9,10,16)(11,13,15)(12,14,17);(1,3,4,7)(2,5,6,8);(1,2)(3,5)(4,6)(7,8)(10,12)(11,14)(13,15)(16,17)Polycyclic presentation:
2^3,3,2,3^2|b^2=e,c^a=ce,f^a=f^2,g^a=g^2,c^b=ce,f^d=fg50 conjugacy classes — view the full character table.
| maximal subgroup of | #11721195, #11721197, #11721199, … |
| maximal normal subgroup of | #11721195, #11721197, #11721199, … |
| maximal quotient of | #64086⋊C3, #64087⋊C3, #64088⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 361);