Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C27 |
| direct product | C2×C7×#953 |
| Fitting | C25×C3×C7 ⋊ C2 |
| Hall | C3×C7 ⋊ C2×#120 |
| Sylow | C3 ⋊ C2×C7×#120 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C7 | C22×D3 |
| commutator, cosocle | C22×C3 | C24×C7 |
| Frattini | C22 | C23×D3×C7 |
| Fitting | C25×C3×C7 | C2 |
| socle | C23×C3×C7 | C23 |
| hypercenter | C25×C7 | D3 |
| hypocenter | C3 | C2×C7×#120 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23×C7 ◁ C25×C7 | |
1324 conjugacy classes of subgroups; only the 318 normal subgroups are shown.
This group has 1324 conjugacy classes of subgroups — too many to draw. Only its 318 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
420 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10845);