Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| isoclinism | isoclinic class of C7⋊SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C3×#41⋊ΓL(1,𝔽8) |
| unambiguous semidirect product | (C3×#41)⋊ΓL(1,𝔽8) |
| Fitting | C3×C7×#41 ⋊ C3 |
| Hall | C7×#41 ⋊ C32 |
| Sylow | #41 ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×#41⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | C22⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7×Q8 | C2×C32 |
| Frattini | C2 | C2×C3×C22⋊ΓL(1,𝔽8) |
| Fitting | C3×C7×#41 | C3 |
| socle | C2×C3×C7 | C2×A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C3×C4 | |
80 conjugacy classes of subgroups.
| transitive degree | μt(G) = 336 |
102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 825);