Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Hall | C7×#585 ⋊ C3 |
| Sylow | #585 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ #5945 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×C22⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7×Q8 | C23×C3 |
| Frattini | C2 | C23×C22⋊ΓL(1,𝔽8) |
| Fitting | C7×#585 | C3 |
| socle | C2×C7 | C23×A4 |
| hypercenter | #41 | C22⋊ΓL(1,𝔽8) |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C4 ◁ #41 | |
390 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 82 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:
2^3,3,2^2,7,2|b^2=h,e^2=h,f^2=h,c^a=ch,e^d=ef,f^d=e,g^d=g^2,f^e=fh106 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11606);