Finite Group DatabasebetaFinite Group Databasebeta| order | 9996 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊Dic119 |
| derived, Hall | C72×C17 ⋊ C3×C4 |
| Hall | C7⋊Dic119 ⋊ C3 |
| Hall | C7×C17×ΓL(1,𝔽8) ⋊ C4 |
| Hall | C7×ΓL(1,𝔽8) ⋊ Dic17 |
| Sylow | C72 ⋊ C3×Dic17 |
| Sylow | C17 ⋊ ΓL(1,𝔽8)⋊Dic7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C17⋊(ΓL(1,𝔽8)⋊D7) |
| commutator, hypocenter | C72×C17 | C3×C4 |
| Fitting, socle, cosocle | C2×C72×C17 | C2×C3 |
| prime radical | C2×C7×C17×ΓL(1,𝔽8) | C2 |
| LCS | G ▷ C72×C17 | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,7^2,17,2|a^2=f,c^a=c^6,d^a=d^6,e^a=e^16,c^b=c^2604 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(9996, 54);