Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | #90015 ⋊ C3 |
| Sylow | C7 ⋊ #458⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #10620804 |
| commutator, Fitting, cosocle, hypocenter | C7×H(5,𝔽2) | C2×C3 |
| Frattini | C23 | D7⋊A4 |
| socle | C2×C7 | #120⋊C3 |
| prime radical | C7⋊#997 | C2 |
| LCS | G ▷ C7×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
90 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2,3,2^4,7,2|a^2=h,c^a=ce,d^a=def,g^a=g^6,c^b=cd,d^b=c,e^b=fh,f^b=efh,g^b=g^2,f^c=fh,e^d=eh,f^d=fh33 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1471);