Finite Group DatabasebetaFinite Group Databasebeta| order | 17724 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ((C211⋊C3)⋊C4)⋊C7 |
| unambiguous semidirect product | ((C211⋊C3)⋊C7)⋊C4 |
| unambiguous semidirect product | (Dic211⋊C7)⋊C3 |
| derived, Sylow | C211 ⋊ C3×C4×C7 |
| Hall | Dic211⋊C7 ⋊ C3 |
| Hall | (C211⋊C3)⋊C7 ⋊ C4 |
| Hall | (C211⋊C3)⋊C4 ⋊ C7 |
| Hall | C211⋊C7 ⋊ C3×C4 |
| Hall | Dic211 ⋊ C3×C7 |
| Hall | C211⋊C3 ⋊ C4×C7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | (D211⋊C3)⋊C7 |
| commutator, hypocenter | C211 | C3×C4×C7 |
| Fitting, socle, cosocle | C2×C211 | C2×C3×C7 |
| prime radical | C2×(C211⋊C3)⋊C7 | C2 |
| LCS | G ▷ C211 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 844 |
Polycyclic presentation:
2,3,7,211,2|a^2=e,d^a=d^210,d^b=d^196,d^c=d^17194 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17724, 30);