Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | #89443 ⋊ C3 |
| Sylow | C7 ⋊ #392⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #10620797 |
| commutator, Fitting, cosocle, hypocenter | C7×#95 | C2×C3 |
| Frattini | C23 | D7⋊A4 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C7×#95 | |
| UCS | C1 ◁ C2 | |
134 conjugacy classes of subgroups.
| transitive degree | μt(G) = 84 |
Polycyclic presentation:
2,3,2^2,7,2^3|a^2=fg,c^2=f,d^2=fh,c^a=cg,d^a=dgh,e^a=e^6,c^b=cd,d^b=cgh,e^b=e^2,f^b=fgh,g^b=h,h^b=gh,d^c=dfg38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1455);