Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D3×C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×D3×C3⋊Dic3 |
| unambiguous semidirect product | (C3×D3)⋊(C3×Dic3) |
| derived | C33 ⋊ C2×C3×C4 |
| Sylow | C34 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D3×C3⋊D3 |
| commutator, hypocenter | C33 | C2×C3×C4 |
| Frattini | C2 | C3×D3×C3⋊D3 |
| Fitting, socle | C2×C34 | C22 |
| prime radical | C2×C33×D3 | C2 |
| cosocle | C2×C33 | C22×C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C3 | |
394 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×#622, C3×#88925, D3×C3⋊#622, … |
| maximal normal subgroup of | C3×D3×#622, C3×#88925, D3×C3⋊#622, … |
| maximal quotient of | C3×D3×C3⋊(C3⋊C8), C3×#88786, C3×Dic3×C3⋊Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 651);