Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C9 · Schur multiplier (H2) = C2 |
| direct product | D4×C27⋊C3 |
| unambiguous semidirect product | C4⋊(C2×C27⋊C3) |
| unambiguous semidirect product | (D4×C27)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C9 | C22×C32 |
| commutator, socle | C2×C3 | C22×C3×C9 |
| prime radical | C32×D4 | C9 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4)(5,6,9,7,11,17,13,20,25,8,12,16,14,21,24,22,28,30,15,10,18,23,19,26,29,27,31);(2,4)(6,10,12)(9,16,18)(11,19,21)(17,24,26)(20,27,28)(25,30,31)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 218);