Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#10621501 |
| Fitting | C22×C7×Q8 ⋊ D3 |
| Sylow | C7 ⋊ C2×#983 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×C7⋊S4 |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C23 |
| Frattini | C2 | C22×C7⋊S4 |
| Fitting | C22×C7×Q8 | D3 |
| socle | C22×C7 | C2×S4 |
| hypercenter | C23 | C7⋊S4 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C23 | |
306 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11405);