Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of C9⋊SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C32 |
| direct product | C3×#6204 |
| Sylow | Q8 ⋊ C3×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C3×A4 |
| commutator, cosocle | C3×Q8 | C33 |
| Frattini | C2×C3 | C32×A4 |
| Fitting | C33×Q8 | C3 |
| prime radical, hypercenter | C2×C33 | A4 |
| hypocenter | Q8 | C3×3−1+2 |
| LCS | G ▷ C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C33 | |
142 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
93 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#3688⋊C3, C3×#3689⋊C3, C3⋊(#3688⋊C3), … |
| maximal normal subgroup of | C3×#3688⋊C3, C3×#3689⋊C3, C3⋊(#3688⋊C3), … |
| maximal quotient of | C2×C3×#6204, C3×(Q8⋊C9)⋊C9, C9×#6204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 437);