Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C3×C7 ⋊ D32 |
| Sylow | C7 ⋊ D96 |
| Sylow | C3 ⋊ #89314 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10620474 |
| commutator, Fitting, cosocle | C3×C7×C16 | C22 |
| Frattini, hypercenter | C16 | D3×D7 |
| socle | C2×C3×C7 | D16 |
| hypocenter | C3×C7 | D32 |
| LCS | G ▷ C3×C7×C16 ▷ C3×C7×C8 ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 672 |
Polycyclic presentation:
2^6,3,7|a^2=c,c^2=de,d^2=ef,e^2=f,b^a=bc,h^a=h^6,c^b=cd,d^b=de,e^b=ef,g^b=g^2132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 137);