Finite Group DatabasebetaFinite Group Databasebeta| order | 9996 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D51⋊D49 |
| unambiguous semidirect product | D147⋊D17 |
| unambiguous semidirect product | D833⋊D3 |
| Fitting, derived, Hall | C3×C17×C49 ⋊ C22 |
| Hall | C17×C49 ⋊ C2×D3 |
| Hall | C3×C49 ⋊ C2×D17 |
| Hall | C3×C17 ⋊ C2×D49 |
| Sylow | C49 ⋊ D3×D17 |
| Sylow | C17 ⋊ D3×D49 |
| Sylow | C3 ⋊ D17×D49 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C17×C49 | C22 |
| Frattini | C7 | D21⋊D17 |
| socle | C3×C7×C17 | C2×D7 |
| LCS | G ▷ C3×C17×C49 | |
| UCS | C1 | |
60 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,7,17,7|d^7=f^6,d^a=d^6f,e^a=e^16,f^a=f^6,c^b=c^2,d^b=d^6f,f^b=f^6678 conjugacy classes — view the full character table.
| maximal subgroup of | D153⋊D49, D51⋊(D49⋊C3), C3×D51⋊D49, … |
| maximal normal subgroup of | D51⋊(D49⋊C3), C3×D51⋊D49 |
| maximal quotient of | D153⋊D49, C3×D51⋊D49, D833⋊(C3⋊D3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(9996, 30);