Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| isoclinism | isoclinic class of C7⋊(C42⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | C7⋊(C42⋊C9) |
| unambiguous semidirect product | C42⋊(C7⋊C9) |
| derived, Hall | C42×C7 ⋊ C9 |
| Sylow | C42 ⋊ C7⋊C9 |
| Sylow | C7 ⋊ C42⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C7⋊(C42⋊C3) |
| commutator, hypocenter | C42×C7 | C9 |
| Frattini | C22×C3 | C22⋊ΓL(1,𝔽8) |
| Fitting, cosocle | C3×C42×C7 | C3 |
| socle, prime radical | C22×C3×C7 | A4 |
| LCS | G ▷ C42×C7 | |
| UCS | C1 ◁ C3 | |
34 conjugacy classes of subgroups.
| transitive degree | μt(G) = 252 |
Polycyclic presentation:
3^2,2^2,7,2^2|a^3=b,c^2=f,d^2=fg,c^a=cd,d^a=cfg,e^a=e^2,f^a=g,g^a=fg120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 57);