Finite Group DatabasebetaFinite Group Databasebeta| order | 9996 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (ΓL(1,𝔽8)⋊D7)⋊D17 |
| unambiguous semidirect product | (C17⋊AGL(1,𝔽7))⋊D7 |
| Fitting, socle, derived, Hall | C72×C17 ⋊ C22×C3 |
| Hall | (C7⋊D7)⋊D17 ⋊ C3 |
| Hall | C7×C17×ΓL(1,𝔽8) ⋊ C22 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C2×D17 |
| Sylow | C72 ⋊ C2×C3×D17 |
| Sylow | C17 ⋊ D7×AGL(1,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C17 | C22×C3 |
| LCS | G ▷ C72×C17 | |
| UCS | C1 | |
84 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,7^2,17|d^a=d^6,f^a=f^16,d^b=d^6,e^b=e^6,d^c=d^2211 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(9996, 96);