Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C36 |
| direct product | C33×SL(2,𝔽3) |
| unambiguous semidirect product | Q8⋊C34 |
| unambiguous semidirect product | (C3×Q8)⋊C33 |
| Fitting | C33×Q8 ⋊ C3 |
| derived, Sylow | Q8 ⋊ C34 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C33 | A4 |
| commutator, cosocle, hypocenter | Q8 | C34 |
| Frattini | C2 | C33×A4 |
| Fitting | C33×Q8 | C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C33 | |
664 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
189 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 678);