Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #6290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C3 |
| direct product | C3×#6293 |
| Sylow | C34 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D3×C3⋊D3 |
| commutator, cosocle | C2×C33 | C22×C3 |
| Frattini | C2 | C3×D3×C3⋊D3 |
| Fitting, socle, prime radical | C2×C34 | C22 |
| hypocenter | C33 | C3×Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2×C3 | |
314 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
99 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×#627, C3×#88957, C3×#88960, … |
| maximal normal subgroup of | C3×D3×#627, C3×#88957, C3×#88960, … |
| maximal quotient of | C3×Dic3⋊(C3⋊Dic3), C3×#88806, C3×#88807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 657);