Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | SL(2,𝔽3)×D21 |
| unambiguous semidirect product | C3⋊(D7×SL(2,𝔽3)) |
| Fitting, derived | C3×C7×Q8 ⋊ C2×C3 |
| Sylow | C7 ⋊ D3×SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×D21 |
| commutator, Fitting, cosocle, hypocenter | C3×C7×Q8 | C2×C3 |
| socle | C2×C3×C7 | C2×A4 |
| LCS | G ▷ C3×C7×Q8 | |
| UCS | C1 ◁ C2 | |
78 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | D21×#997, SL(2,𝔽3)×C3⋊AGL(1,𝔽7), D3×D7×SL(2,𝔽3) |
| maximal normal subgroup of | SL(2,𝔽3)×C3⋊AGL(1,𝔽7), D3×D7×SL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 677);