Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C2×C7×GL(2,ℤ/4) |
| Fitting | C25×C7 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C7 | C2×S4 |
| commutator, cosocle | C2×A4 | C23×C7 |
| Frattini | C2 | C22×C7×S4 |
| Fitting | C25×C7 | D3 |
| socle | C24×C7 | C2×D3 |
| hypercenter | C23×C7 | S4 |
| hypocenter | A4 | C2×C7×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 | |
542 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
196 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C7×A4⋊S4, C2×ΓL(1,𝔽8)×GL(2,ℤ/4), C2×D7×GL(2,ℤ/4) |
| maximal normal subgroup of | C2×ΓL(1,𝔽8)×GL(2,ℤ/4), C2×D7×GL(2,ℤ/4) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11394);