Finite Group DatabasebetaFinite Group Databasebeta| order | 9996 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C7 |
| unambiguous semidirect product | C3⋊(C7⋊(C7⋊#595)) |
| unambiguous semidirect product | C7⋊(C3⋊(C7⋊#595)) |
| Fitting, socle, derived, Hall | C3×C72×C17 ⋊ C4 |
| Hall | C72×C17 ⋊ Dic3 |
| Hall | C3×C72 ⋊ #595 |
| Hall | C3×C17 ⋊ C7⋊Dic7 |
| Sylow | C72 ⋊ C3⋊#595 |
| Sylow | C17 ⋊ C3⋊(C7⋊Dic7) |
| Sylow | C3 ⋊ C7⋊(C7⋊#595) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C72×C17 | C4 |
| prime radical, cosocle | C3×C72×D17 | C2 |
| LCS | G ▷ C3×C72×C17 | |
| UCS | C1 | |
120 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,7^2,17|a^2=b,c^a=c^2,d^a=d^6,e^a=e^6,f^a=f^13,f^b=f^16738 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(9996, 152);