Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C33 |
| direct product | C33×D12 |
| unambiguous semidirect product | C4⋊(C33×D3) |
| Fitting | C34×C4 ⋊ C2 |
| Sylow | C34 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C33 | C2×D3 |
| commutator, cosocle | C2×C3 | C22×C33 |
| Frattini | C2 | C2×C33×D3 |
| Fitting | C34×C4 | C2 |
| socle | C2×C34 | C22 |
| hypercenter | C33×C4 | D3 |
| hypocenter | C3 | C33×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C33 ◁ C33×C4 | |
682 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
243 conjugacy classes — view the full character table.
| maximal subgroup of | C32×#3713, C32×#3714, C32×#3715, … |
| maximal normal subgroup of | C32×#3713, C32×#3714, C32×#3715, … |
| maximal quotient of | C33×#204, C33×D24, C33×Dic12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 681);