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) = C22 |
| unambiguous semidirect product | (C2×AGL(1,ℤ/9))⋊C4 |
| unambiguous semidirect product | ((C2×D9)⋊C4)⋊C3 |
| Fitting | C2×C4×3−1+2 ⋊ C2 |
| Sylow | 3−1+2 ⋊ C22⋊C4 |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
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|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 | #63434⋊C3, C4×C4⋊AGL(1,ℤ/9), #63437⋊C3, … |
| maximal normal subgroup of | #63434⋊C3, C4×C4⋊AGL(1,ℤ/9), #63437⋊C3, … |
| maximal quotient of | #63364⋊C3, #63365⋊C3, #63368⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 147);