Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C32 |
| Fitting | C22×C33 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×D3×A4 |
| commutator, socle, cosocle | C22×C32 | C2×C32 |
| Fitting | C22×C33 | C2×C3 |
| hypercenter | C32 | D3×A4 |
| hypocenter | C22×C3 | C2×H(3,𝔽3) |
| LCS | G ▷ C22×C32 ▷ C22×C3 | |
| UCS | C1 ◁ C3 ◁ C32 | |
178 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 30 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 | D3×(C3⋊S4)⋊C3, D3×#6258, D3×(C3⋊D3)⋊A4, … |
| maximal normal subgroup of | D3×(C3⋊S4)⋊C3, D3×#6258, D3×(C3⋊D3)⋊A4, … |
| maximal quotient of | C3⋊(#41⋊H(3,𝔽3)), D3×Q8⋊H(3,𝔽3), Dic3×C22⋊H(3,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 587);