Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | ΓL(1,𝔽8)×#41⋊C3 |
| unambiguous semidirect product | (#41×ΓL(1,𝔽8))⋊C3 |
| unambiguous semidirect product | (C7×#41⋊C3)⋊C3 |
| unambiguous semidirect product | (#41⋊ΓL(1,𝔽8))⋊C3 |
| Fitting, Hall | C7×#41 ⋊ C32 |
| Sylow | #41 ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×#41⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | A4×ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7×Q8 | C2×C32 |
| Frattini | C2 | C2×A4×ΓL(1,𝔽8) |
| Fitting | C7×#41 | C32 |
| socle | C2×C7 | C2×C3×A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C4 | |
80 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#41⋊C3, ΓL(1,𝔽8)×#985, #72218832, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#41⋊C3, ΓL(1,𝔽8)×#985, #72218832, … |
| maximal quotient of | C4×ΓL(1,𝔽8)×SL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 556);