Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×3−1+2 ⋊ C2 |
| Sylow | 3−1+2 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×AGL(1,ℤ/9) |
| commutator, cosocle | C2×C9 | C23×C3 |
| Frattini, socle | C2×C3 | C22×C3×D3 |
| Fitting | Q8×3−1+2 | C2 |
| hypercenter | Q8 | AGL(1,ℤ/9) |
| hypocenter | C9 | C3×#41 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,5,6,7,2,8,3,9,4)(10,12,13,16)(11,14,15,17);(10,11,13,15)(12,17,16,14);(2,4)(3,7)(5,6)(8,9)(12,16)(14,17)Polycyclic presentation:
2^3,3,2,3^2|b^2=e,c^2=e,f^3=g^2,c^a=ce,f^a=f^2g,g^a=g^2,c^b=ce,f^d=fg50 conjugacy classes — view the full character table.
| maximal subgroup of | #63476⋊C3, #63478⋊C3, #63480⋊C3, … |
| maximal normal subgroup of | #63476⋊C3, #63478⋊C3, #63480⋊C3, … |
| maximal quotient of | #63452⋊C3, #63454⋊C3, #63455⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 371);