Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | D7⋊(C22⋊SL(2,𝔽3)) |
| Fitting, derived | C22×C7×Q8 ⋊ C2×C3 |
| Hall | C22×Q8×D7 ⋊ C3 |
| Sylow | C7 ⋊ C2×C22⋊SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7⋊(C22⋊A4) |
| commutator, Fitting, cosocle, hypocenter | C22×C7×Q8 | C2×C3 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C22×C7×Q8 | |
| UCS | C1 ◁ C2 | |
350 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 31 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:
2,3,2^4,7,2|e^2=h,f^2=h,g^a=g^6,c^b=cd,d^b=c,e^b=ef,f^b=e,g^b=g^4,f^e=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11498);