Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C22×#66686 |
| Hall | C3×C7 ⋊ C22×#41 |
| Sylow | C7 ⋊ C22×#235 |
| Sylow | C3 ⋊ C22×#1159 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×D3×D7 |
| commutator, cosocle | C2×C3×C7 | C25 |
| Frattini | C2 | C23×D3×D7 |
| Fitting | C24×C3×C7 | C22 |
| socle | C23×C3×C7 | C23 |
| hypercenter | C24 | D3×D7 |
| hypocenter | C3×C7 | C22×#41 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C23 ◁ C24 | |
1780 conjugacy classes of subgroups; only the 568 normal subgroups are shown.
This group has 1780 conjugacy classes of subgroups — too many to draw. Only its 568 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 672 |
180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11572);