Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | D3×#6500 |
| Fitting | C2×C3×C7×D4 ⋊ C22 |
| Hall | C3×C7 ⋊ C2×#120 |
| Sylow | C7 ⋊ D3×#120 |
| Sylow | C3 ⋊ C2×#6500 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3×D7 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C2×C3×C7×D4 | C22 |
| hypercenter | C2×D4 | D3×D7 |
| hypocenter | C3×C7 | C2×#120 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
1324 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
132 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×#120, D3×#6500⋊C3, D3×#90016, … |
| maximal normal subgroup of | D3×D7×#120, D3×#6500⋊C3, D3×#90016, … |
| maximal quotient of | D3×#89804, D3×#89849, D3×#89970, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 9569);