Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | Dic3×#272 |
| derived, Sylow | C3×H(3,𝔽3) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | D3×C3⋊D3 |
| commutator, hypocenter | C3×H(3,𝔽3) | C2×C4 |
| Fitting, cosocle | C2×C3×H(3,𝔽3) | C22 |
| socle | C2×C32 | C2×C3⋊D3 |
| prime radical | C2×C3×#272 | C2 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
237 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#1088, Dic3⋊#1088, #13048322, … |
| maximal normal subgroup of | Dic3×#1088, Dic3⋊#1088, #13048322, … |
| maximal quotient of | C3⋊C8×#272, #13047689, Dic3×#1084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 408);