Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×C7×#123 ⋊ C2 |
| Hall | C3×C7 ⋊ #536 |
| Sylow | C7 ⋊ #5648 |
| Sylow | C3 ⋊ #90339 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D21 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C7×#123 | C2 |
| prime radical | C2×#66730 | C2 |
| hypercenter | #123 | D21 |
| hypocenter | C3×C7 | #536 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ #123 | |
540 conjugacy classes of subgroups.
| transitive degree | μt(G) = 672 |
Polycyclic presentation:
2^6,3,7|a^2=f,d^2=f,d^a=df,g^a=g^2,h^a=h^6,c^b=cf,d^c=de162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11028);