Finite Group DatabasebetaFinite Group Databasebeta| order | 17724 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #22579723⋊C2 |
| unambiguous semidirect product | #22579727⋊C2 |
| unambiguous semidirect product | #22579731⋊C2 |
| Fitting, socle, derived, Hall | C7×C211 ⋊ C22×C3 |
| Hall | D7×D211 ⋊ C3 |
| Hall | #17984522 ⋊ C22 |
| Sylow | C211 ⋊ C2×AGL(1,𝔽7) |
| Sylow | C7 ⋊ C2×D211⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C211 | C22×C3 |
| LCS | G ▷ C7×C211 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1477 |
Polycyclic presentation:
2^2,3,7,211|e^a=e^210,d^b=d^6,d^c=d^2,e^c=e^196189 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17724, 41);