Finite Group DatabasebetaFinite Group Databasebeta| order | 816 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C17 · Schur multiplier (H2) = C1 |
| direct product | C2×C17×SL(2,𝔽3) |
| unambiguous semidirect product | (C2×Q8)⋊(C3×C17) |
| unambiguous semidirect product | (C2×Q8×C17)⋊C3 |
| Fitting, Hall | C2×Q8×C17 ⋊ C3 |
| derived | Q8 ⋊ C2×C3×C17 |
| Sylow | C2×Q8 ⋊ C3×C17 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C17 | A4 |
| commutator, cosocle, hypocenter | Q8 | C2×C3×C17 |
| Frattini | C2 | C2×A4×C17 |
| Fitting | C2×Q8×C17 | C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C17 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 272 |
238 conjugacy classes — view the full character table.
| maximal subgroup of | C17×#859, #13108921, SL(2,𝔽3)×Dic17, … |
| maximal normal subgroup of | C17×#859, #13108921, SL(2,𝔽3)×Dic17, … |
| maximal quotient of | C4×C17×SL(2,𝔽3), C22×C17×SL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(816, 149);