Finite Group DatabasebetaFinite Group Databasebeta| order | 17724 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C7 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊((C211⋊C3)⋊C7) |
| unambiguous semidirect product | (C211⋊C7)⋊A4 |
| unambiguous semidirect product | (C22⋊(C211⋊C3))⋊C7 |
| Fitting, socle, derived, Hall | C22×C211 ⋊ C3×C7 |
| Hall | C22×C211⋊C7 ⋊ C3 |
| Hall | C22⋊(C211⋊C3) ⋊ C7 |
| Hall | C211⋊C7 ⋊ A4 |
| Sylow | C211 ⋊ C7×A4 |
| Sylow | C22 ⋊ (C211⋊C3)⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C211 | C3×C7 |
| LCS | G ▷ C22×C211 | |
| UCS | C1 | |
20 conjugacy classes of subgroups.
| transitive degree | μt(G) = 844 |
Polycyclic presentation:
3,7,2^2,211|c^a=d,d^a=cd,e^a=e^14,e^b=e^17168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17724, 35);