Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C2×ΓL(1,𝔽8)×SL(2,𝔽3) |
| unambiguous semidirect product | (C2×Q8×ΓL(1,𝔽8))⋊C3 |
| unambiguous semidirect product | (C2×C7×SL(2,𝔽3))⋊C3 |
| unambiguous semidirect product | (C2×C7⋊SL(2,𝔽3))⋊C3 |
| Fitting, Hall | C2×C7×Q8 ⋊ C32 |
| derived | C7×Q8 ⋊ C2×C32 |
| Sylow | C2×Q8 ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C2×C3×SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A4×ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7×Q8 | C2×C32 |
| Frattini | C2 | C2×A4×ΓL(1,𝔽8) |
| Fitting | C2×C7×Q8 | C32 |
| socle | C22×C7 | C3×A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C22 | |
100 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
70 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 555);