Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C22⋊D7)⋊SL(2,𝔽3) |
| Fitting | C22×C7×Q8 ⋊ C2×C3 |
| Hall | Q8×C22⋊D7 ⋊ C3 |
| Sylow | C7 ⋊ D4×SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D7⋊A4 |
| commutator, cosocle | C2×C7×Q8 | C22×C3 |
| Fitting | C22×C7×Q8 | C2×C3 |
| socle | C22×C7 | C22×A4 |
| hypercenter | C23 | D7⋊A4 |
| hypocenter | C7×Q8 | C3×D4 |
| LCS | G ▷ C2×C7×Q8 ▷ C7×Q8 | |
| UCS | C1 ◁ C22 ◁ C23 | |
210 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^2,3,2^3,7,2|e^2=h,f^2=h,b^a=bd,g^a=g^6,e^c=ef,f^c=e,g^c=g^2,f^e=fh55 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6964);