Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | D4×AΣL(1,𝔽27) |
| unambiguous semidirect product | C4⋊(C2×AΣL(1,𝔽27)) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×H(3,𝔽3) |
| commutator, Frattini, cosocle | C2×C32 | C22×C32 |
| LCS | G ▷ C2×C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C32×D4 ◁ G | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,3,4)(5,6,8)(7,9,12)(10,13,11);(2,4)(6,9,10)85 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 219);