Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C3⋊#226 |
| automorphism | Aut(G) = C22×((C3⋊S4)⋊C3)⋊C2 (order 1728) · Inn(G) = C3⋊S4 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
| direct product | C3×C3⋊#226 |
| unambiguous semidirect product | C3⋊(C3×#226) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C2×C3 |
| Frattini | C2 | C3×C3⋊S4 |
| Fitting | C32×Q8 | D3 |
| socle | C2×C32 | S4 |
| prime radical | C32×SL(2,𝔽3) | C2 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
97 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,15,10,7,5,6,12,13)(2,3,14,9,8,11,16,4)(17,18,19)(21,22);(20,21,22);(3,9,12)(4,10,11)(6,13,16)(7,14,15)45 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#64196, C3×#64197, C3×D3×#226, … |
| maximal normal subgroup of | C3×#64196, C3×#64197, C3×D3×#226, … |
| maximal quotient of | C3×C3⋊#859, C2×C3×C3⋊#226, C9×C3⋊#226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 616);