Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C32 |
| direct product | #67×H(3,𝔽3) |
| unambiguous semidirect product | (C32×#67)⋊C3 |
| Fitting | C22×C3×H(3,𝔽3) ⋊ C2 |
| Sylow | C3×H(3,𝔽3) ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C3 | C2×C32×D3 |
| commutator, socle, cosocle | C2×C32 | C22×C32 |
| Fitting | C22×C3×H(3,𝔽3) | C2 |
| hypercenter | C22×H(3,𝔽3) | D3 |
| hypocenter | C3 | D4×H(3,𝔽3) |
| LCS | G ▷ C2×C32 ▷ C3 | |
| UCS | C1 ◁ C2×C3 ◁ C22×H(3,𝔽3) | |
261 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
99 conjugacy classes — view the full character table.
| maximal subgroup of | #89029⋊C3, #89028⋊C3, #67×(C3⋊D3)⋊C3, … |
| maximal normal subgroup of | #89029⋊C3, #89028⋊C3, #67×(C3⋊D3)⋊C3, … |
| maximal quotient of | H(3,𝔽3)×Dic3⋊C4, H(3,𝔽3)×(C2×D3)⋊C4, H(3,𝔽3)×#213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 464);