Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C4 · Schur multiplier (H2) = C2×C33 |
| direct product | C2×C33×Dic3 |
| unambiguous semidirect product | (C2×C3)⋊(C33×C4) |
| derived | C3 ⋊ C2×C33×C4 |
| Sylow | C34 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C33 | D3 |
| commutator, hypocenter | C3 | C2×C33×C4 |
| Frattini | C2 | C2×C33×D3 |
| Fitting, socle, prime radical | C22×C34 | C2 |
| cosocle | C2×C3 | C22×C33 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C22×C33 | |
838 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
324 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C32×Dic32, C32×(C2×D3)⋊Dic3, C32×#3722, … |
| maximal normal subgroup of | C32×Dic32, C32×(C2×D3)⋊Dic3, C32×#3722, … |
| maximal quotient of | C2×C33×C3⋊C8, C33×#208, C33×C4×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 682);