Finite Group DatabasebetaFinite Group Databasebeta| order | 882 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C7 · Schur multiplier (H2) = C3 |
| direct product | C2×C3×C7×ΓL(1,𝔽8) |
| unambiguous semidirect product | (C2×C7)⋊(C32×C7) |
| unambiguous semidirect product | (C2×C3×C7)⋊(C3×C7) |
| Fitting, socle | C2×C3×C72 ⋊ C3 |
| derived | C7 ⋊ C2×C32×C7 |
| Hall | C2×C72 ⋊ C32 |
| Sylow | C72 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C7 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C2×C32×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C72 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3×C7 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 126 |
210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(882, 57);