Finite Group DatabasebetaFinite Group Databasebeta| order | 9996 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C7 |
| unambiguous semidirect product | C3⋊#16809726 |
| Fitting, socle, derived, Hall | C3×C72×C17 ⋊ C4 |
| Hall | C72×C17 ⋊ Dic3 |
| Hall | C3×C72 ⋊ Dic17 |
| Hall | C3×C17 ⋊ #5997 |
| Sylow | C72 ⋊ Dic51 |
| Sylow | C17 ⋊ C3⋊#5997 |
| Sylow | C3 ⋊ #16809726 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C72×C17 | C4 |
| prime radical, cosocle | C3×C17×C7⋊D7 | C2 |
| LCS | G ▷ C3×C72×C17 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,7^2,17|a^2=b,c^a=c^2,d^a=d^4e^6,e^a=d^3e^3,f^a=f^16,d^b=d^6,e^b=e^6666 conjugacy classes — view the full character table.
| maximal subgroup of | #44472558, C9⋊#16809726, C3⋊#23756207, … |
| maximal normal subgroup of | C3⋊(#16809726⋊C3), C3×C3⋊#16809726 |
| maximal quotient of | C9⋊#16809726, C3×C3⋊#16809726, C3⋊(C3⋊#16809726) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(9996, 167);