Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C7⋊#5932 |
| Sylow | C7 ⋊ #5932 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C7⋊#1020 |
| commutator, prime radical, cosocle, hypocenter | C7×#997 | C2 |
| Fitting | C7×H(5,𝔽2) | D3 |
| socle | C2×C7 | #1020 |
| LCS | G ▷ C7×#997 | |
| UCS | C1 ◁ C2 | |
142 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2,3,7,2^5|a^2=h,b^a=b^2,c^a=c^6,d^a=de,f^a=fg,d^b=e,e^b=de,f^b=g,g^b=fg,g^d=gh,f^e=fh46 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11425);