Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | A4×#272 |
| Fitting, derived | C22×H(3,𝔽3) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | A4×C3⋊D3 |
| commutator, Fitting, cosocle, hypocenter | C22×H(3,𝔽3) | C2×C3 |
| socle | C22×C3 | C3×C3⋊D3 |
| LCS | G ▷ C22×H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
196 conjugacy classes of subgroups.
| transitive degree | μt(G) = 36 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | A4⋊#1088, A4×#1088, S4×#272, … |
| maximal normal subgroup of | A4⋊#1088, A4×#1088, S4×#272, … |
| maximal quotient of | #13048464, SL(2,𝔽3)×#272, A4×#1084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 592);