Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C3×C7 ⋊ Q64 |
| Sylow | C7 ⋊ #4522 |
| Sylow | C3 ⋊ Dic112 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10620473 |
| commutator, Fitting, cosocle | C3×C7×C16 | C22 |
| Frattini, hypercenter | C16 | D3×D7 |
| socle, prime radical | C2×C3×C7 | D16 |
| hypocenter | C3×C7 | Q64 |
| LCS | G ▷ C3×C7×C16 ▷ C3×C7×C8 ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 74 |
| transitive degree | μt(G) = 1344 |
Polycyclic presentation:
2^6,3,7|a^2=f,b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,h^a=h^6,g^b=g^2174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 145);