Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C3⋊(#41⋊C3) |
| automorphism | Aut(G) = C2×C3×D3×S4 (order 864) · Inn(G) = D3×A4 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| direct product | D3×Q8⋊C9 |
| unambiguous semidirect product | C3⋊(C2×Q8⋊C9) |
| unambiguous semidirect product | (D3×Q8)⋊C9 |
| derived | C3×Q8 ⋊ C2×C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | D3×A4 |
| commutator, hypocenter | C3×Q8 | C2×C9 |
| Fitting, cosocle | C32×Q8 | C2×C3 |
| socle | C2×C32 | C2×A4 |
| prime radical | C2×C3×D3 | A4 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C2×C3 | |
65 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,3,5)(6,8,7)(10,11)(12,16,20,14,19,15,17,13,18);(1,7,4,5)(2,3,6,8)(9,10,11)63 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3688, #11722250, #11722251, … |
| maximal normal subgroup of | D3×#3688, #11722250, #11722251, … |
| maximal quotient of | Dic3×Q8⋊C9, C2×D3×Q8⋊C9, D3×Q8⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 268);