Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C7⋊#226) |
| unambiguous semidirect product | C7⋊(C3⋊#226) |
| Sylow | C7 ⋊ C3⋊#226 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊(C7⋊S4) |
| commutator, prime radical, cosocle, hypocenter | C3×C7×SL(2,𝔽3) | C2 |
| Fitting | C3×C7×Q8 | D3 |
| socle | C2×C3×C7 | S4 |
| LCS | G ▷ C3×C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 336 |
Polycyclic presentation:
2,3^2,7,2^3|a^2=g,e^2=g,f^2=g,b^a=b^2,c^a=c^2,d^a=d^6,e^a=ef,f^a=fg,e^b=f,f^b=efg,f^e=fg78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 634);