Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of D7⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C16 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C22⋊ΓL(1,𝔽8))⋊C16 |
| unambiguous semidirect product | (C7⋊C16)⋊A4 |
| derived | C22×C7 ⋊ C3×C16 |
| Hall | C22×C7⋊C16 ⋊ C3 |
| Sylow | C7 ⋊ A4×C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | D7⋊A4 |
| commutator, hypocenter | C22×C7 | C3×C16 |
| Fitting, cosocle | C22×C7×C8 | C2×C3 |
| socle | C23×C7 | C3×C8 |
| prime radical | C2×C22⋊ΓL(1,𝔽8) | C8 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C8 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 336 |
Polycyclic presentation:
2,3,2^5,7|a^2=c,c^2=d,d^2=e,h^a=h^6,f^b=fg,g^b=f,h^b=h^296 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1467);