Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of 3−1+2⋊Q8 |
| automorphism | Aut(G) = C24×AGL(1,ℤ/9) (order 864) · Inn(G) = C2×AGL(1,ℤ/9) · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (3−1+2⋊C4)⋊C4 |
| unambiguous semidirect product | (Dic9⋊C4)⋊C3 |
| Sylow | 3−1+2 ⋊ C4⋊C4 |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,5,6,7,2,8,3,9,4)(10,11,12,13)(15,17);(2,4)(3,7)(5,6)(8,9)(14,15,16,17)Polycyclic presentation:
2^2,3,2^2,3^2|a^2=e,b^2=d,f^3=g^2,b^a=be,f^a=f^2g,g^a=g^2,f^c=fg62 conjugacy classes — view the full character table.
| maximal subgroup of | C4×3−1+2⋊Q8, #63432⋊C3, #63434⋊C3, … |
| maximal normal subgroup of | C4×3−1+2⋊Q8, #63432⋊C3, #63434⋊C3, … |
| maximal quotient of | #63366⋊C3, #63367⋊C3, (3−1+2⋊C4)⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 145);