Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | (C2×D7)⋊(C3⋊Dic3) |
| Hall | C32×C7 ⋊ C22⋊C4 |
| Sylow | C32 ⋊ (C2×D7)⋊C4 |
| Sylow | C7 ⋊ #3757 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D7×C3⋊D3 |
| commutator | C2×C32×C7 | C2×C4 |
| Fitting, socle, cosocle | C22×C32×C7 | C22 |
| prime radical | C22×C32×D7 | C2 |
| hypocenter | C32×C7 | C22⋊C4 |
| LCS | G ▷ C2×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C22 | |
204 conjugacy classes of subgroups.
| transitive degree | μt(G) = 504 |
Polycyclic presentation:
2^4,3^2,7|b^2=c,b^a=bd,g^a=g^6,e^b=e^2,f^b=f^2114 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#3757, C3⋊D3×(C2×D7)⋊C4, #72246732, … |
| maximal normal subgroup of | D7×#3757, C3⋊D3×(C2×D7)⋊C4, #72246732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 337);