Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#41⋊ΓL(1,𝔽8)) |
| unambiguous semidirect product | #239⋊ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3⋊(#41⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×C22⋊ΓL(1,𝔽8) |
| commutator, Fitting, cosocle, hypocenter | C3×C7×Q8 | C2×C3 |
| socle | C2×C3×C7 | C2×A4 |
| LCS | G ▷ C3×C7×Q8 | |
| UCS | C1 ◁ C2 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
Polycyclic presentation:
2,3,2^2,3,7,2|a^2=g,c^2=g,d^2=g,e^a=e^2,c^b=cd,d^b=c,f^b=f^4,d^c=dg51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 679);