Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of C3⋊(C3⋊(C3⋊D3)) |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C36 |
| direct product | C22×C3⋊(C3⋊(C3⋊D3)) |
| Fitting, socle | C22×C34 ⋊ C2 |
| derived, Sylow | C34 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊(C3⋊(C3⋊D3)) |
| commutator, cosocle, hypocenter | C34 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C22×C34 | C2 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C22 | |
3392 conjugacy classes of subgroups.
| transitive degree | μt(G) = 324 |
168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 756);