Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | Aut(G) = C23×((C3⋊D3)⋊C3)⋊C2 (order 864) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2×C3 |
| direct product | C8×(C3⋊D3)⋊C3 |
| unambiguous semidirect product | (C8×C3⋊D3)⋊C3 |
| Fitting | C8×H(3,𝔽3) ⋊ C2 |
| derived | C32 ⋊ C2×C3×C8 |
| Sylow | H(3,𝔽3) ⋊ C2×C8 |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(10,15,17,12,13,16)(11,14);(1,2,3,5,4,6,7,8)(9,10,12)(11,13,16)(14,17,15)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 115);