Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (C2×S4)⋊Dic7 |
| Sylow | C7 ⋊ (C2×S4)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D7×S4 |
| commutator | C2×C7×A4 | C2×C4 |
| Fitting, socle | C24×C7 | C2×D3 |
| prime radical | C22×C7×S4 | C2 |
| cosocle | C22×C7×A4 | C22 |
| hypocenter | C7×A4 | C22⋊C4 |
| LCS | G ▷ C2×C7×A4 ▷ C7×A4 | |
| UCS | C1 ◁ C22 | |
286 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2^4,3,7,2^2|a^2=d,b^a=bc,f^a=f^6,e^b=e^2,g^b=gh,g^e=h,h^e=gh88 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6389);