Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of 3−1+2⋊Q8 |
| automorphism | Aut(G) = #120×AGL(1,ℤ/9) (order 1728) · Inn(G) = C2×AGL(1,ℤ/9) · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊(3−1+2⋊C4) |
| unambiguous semidirect product | (C4⋊Dic9)⋊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)(14,15,16,17);(2,4)(3,7)(5,6)(8,9)(10,11,12,13)(15,17)Polycyclic presentation:
2^2,3,2^2,3^2|a^2=d,b^2=e,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 | #63366⋊C3, #63367⋊C3, #63376⋊C3, … |
| maximal normal subgroup of | #63366⋊C3, #63367⋊C3, #63376⋊C3, … |
| maximal quotient of | C4⋊(3−1+2⋊C8), #63372⋊C3, #63377⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 146);