Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #6258 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#6258 |
| Fitting | C22×C33 ⋊ D3 |
| derived | C22⋊H(3,𝔽3) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C3⋊S4 |
| commutator, cosocle, hypocenter | C22⋊H(3,𝔽3) | C2×C3 |
| Frattini | C3 | C3×C3⋊S4 |
| Fitting | C22×C33 | D3 |
| socle | C22×C32 | C3⋊D3 |
| LCS | G ▷ C22⋊H(3,𝔽3) | |
| UCS | C1 ◁ C32 | |
196 conjugacy classes of subgroups.
| transitive degree | μt(G) = 54 |
57 conjugacy classes — view the full character table.
| maximal subgroup of | C3×((C3⋊S4)⋊C3)⋊C2, D3×#6258, #13049222, … |
| maximal normal subgroup of | C3×((C3⋊S4)⋊C3)⋊C2, D3×#6258, #13049222, … |
| maximal quotient of | C3×#88610, C3×#88611, C3×#88613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 572);