Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1 |
| direct product | D3×C7×SL(2,𝔽3) |
| unambiguous semidirect product | C3⋊(C2×C7×SL(2,𝔽3)) |
| unambiguous semidirect product | (D3×Q8)⋊(C3×C7) |
| Fitting | C3×C7×Q8 ⋊ C2×C3 |
| derived | C3×Q8 ⋊ C2×C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D3×A4 |
| commutator, cosocle, hypocenter | C3×Q8 | C2×C3×C7 |
| Frattini | C2 | D3×C7×A4 |
| Fitting | C3×C7×Q8 | C2×C3 |
| socle | C2×C3×C7 | C2×A4 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C2×C7 | |
78 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
147 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C7×#997, D3×ΓL(1,𝔽8)×SL(2,𝔽3), D3×D7×SL(2,𝔽3) |
| maximal normal subgroup of | D3×ΓL(1,𝔽8)×SL(2,𝔽3), D3×D7×SL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 674);