Finite Group DatabasebetaFinite Group Databasebeta| order | 17724 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊#17984521 |
| Fitting, socle, derived, Hall | C22×C7×C211 ⋊ C3 |
| Hall | C7×C211 ⋊ A4 |
| Hall | C22×C211 ⋊ ΓL(1,𝔽8) |
| Sylow | C211 ⋊ C22⋊ΓL(1,𝔽8) |
| Hall | C22×C7 ⋊ C211⋊C3 |
| Sylow | C7 ⋊ C22⋊(C211⋊C3) |
| Sylow | C22 ⋊ #17984521 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C7×C211 | C3 |
| LCS | G ▷ C22×C7×C211 | |
| UCS | C1 | |
20 conjugacy classes of subgroups.
| transitive degree | μt(G) = 5908 |
Polycyclic presentation:
3,2^2,7,211|b^a=c,c^a=bc,d^a=d^2,e^a=e^141972 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17724, 64);