Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | D7×GL(2,ℤ/4) |
| Fitting | C24×C7 ⋊ C2×D3 |
| Sylow | C7 ⋊ C2×GL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7×S4 |
| commutator, cosocle | C2×C7×A4 | C23 |
| Fitting | C24×C7 | C2×D3 |
| socle | C23×C7 | C22×D3 |
| hypercenter | C22 | D7×S4 |
| hypocenter | C7×A4 | C2×D4 |
| LCS | G ▷ C2×C7×A4 ▷ C7×A4 | |
| UCS | C1 ◁ C2 ◁ C22 | |
542 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 542 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) = 84 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | D7×A4⋊S4, AGL(1,𝔽7)×GL(2,ℤ/4), D4×D7×S4, … |
| maximal normal subgroup of | AGL(1,𝔽7)×GL(2,ℤ/4), D4×D7×S4, C2×D7×GL(2,ℤ/4) |
| maximal quotient of | C2×D7×GL(2,ℤ/4), D7×#5429, D7×#87524 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11368);