Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×C22⋊(C7⋊SL(2,𝔽3)) |
| unambiguous semidirect product | (C2×C7)⋊(C22⋊SL(2,𝔽3)) |
| Fitting, Hall | C23×C7×Q8 ⋊ C3 |
| derived | C22×C7×Q8 ⋊ C2×C3 |
| Sylow | C23×Q8 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C2×C22⋊SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C22⋊(C22⋊ΓL(1,𝔽8)) |
| commutator, cosocle, hypocenter | C22×C7×Q8 | C2×C3 |
| Frattini | C2 | C2×C22⋊(C22⋊ΓL(1,𝔽8)) |
| Fitting | C23×C7×Q8 | C3 |
| socle | C24×C7 | A4 |
| LCS | G ▷ C22×C7×Q8 | |
| UCS | C1 ◁ C22 | |
350 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11613);