Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C32 |
| direct product | Q16×H(3,𝔽3) |
| unambiguous semidirect product | (C32×Q16)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×D4 |
| commutator, Frattini, cosocle | C3×C4 | C22×C32 |
| prime radical | C2×H(3,𝔽3) | D4 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C4×H(3,𝔽3) ◁ G | |
99 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,14,17)(12,18,20,25)(13,21,23,16)(15,22,24,19);(2,5,6)(4,8,9)(10,12,14,20)(11,15,17,24)(13,22,23,19)(16,25,21,18)77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 222);