Finite Group DatabasebetaFinite Group Databasebeta| order | 17724 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×#22579727 |
| Fitting, socle | C2×C7×C211 ⋊ C2×C3 |
| derived, Hall | C7×C211 ⋊ C22×C3 |
| Hall | C2×D7×C211 ⋊ C3 |
| Hall | #17984522 ⋊ C22 |
| Sylow | C211 ⋊ C2×AGL(1,𝔽7) |
| Hall | C2×D7 ⋊ C211⋊C3 |
| Sylow | C7 ⋊ C22×C211⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #22579727 |
| commutator, cosocle, hypocenter | C7×C211 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C211 | C2×C3 |
| LCS | G ▷ C7×C211 | |
| UCS | C1 ◁ C2 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2954 |
714 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17724, 50);