Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of Dic36⋊C3 |
| automorphism | Aut(G) = AGL(1,ℤ/8)×AGL(1,ℤ/9) (order 1728) · Inn(G) = C4⋊AGL(1,ℤ/9) · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D72⋊C3 |
| Fitting | C8×3−1+2 ⋊ C2 |
| Sylow | 3−1+2 ⋊ D8 |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,4,7,9,6,3,5,8)(10,11,15,17,13,16,12,14);(2,4)(3,7)(5,6)(8,9)(11,14)(12,15)(16,17)Polycyclic presentation:
2^2,3,2^2,3^2|b^2=de,d^2=e,f^3=g^2,b^a=bd,d^a=de,f^a=f^2g,g^a=g^2,f^c=fg53 conjugacy classes — view the full character table.
| maximal subgroup of | D144⋊C3, #63359⋊C3, #63385⋊C3, … |
| maximal normal subgroup of | D144⋊C3, #63359⋊C3, #63385⋊C3, … |
| maximal quotient of | D144⋊C3, #63359⋊C3, Dic72⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 123);