Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C32 |
| direct product | C2×Q8×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C32 |
| commutator, Frattini, cosocle | C2×C3 | C23×C32 |
| prime radical | C22×H(3,𝔽3) | C22 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
209 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(12,14,17)(15,19,18);(3,4,6,8)(5,9,10,7)(11,12,15)(13,14,18)(16,17,19);(1,2)(3,5,6,10)(4,7,8,9)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 407);