Finite Group DatabasebetaFinite Group Databasebeta| order | 9996 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C7 |
No aliases for this group.
| Fitting, socle, derived, Hall | C72×C17 ⋊ C3×C4 |
| Hall | C7⋊(C7⋊#595) ⋊ C3 |
| Hall | C17×#3862 ⋊ C4 |
| Hall | #3862 ⋊ #595 |
| Sylow | C72 ⋊ C3×#595 |
| Sylow | C17 ⋊ #10596875 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C72×C17 | C3×C4 |
| prime radical | D17×#3862 | C2 |
| cosocle | C72×D17 | C2×C3 |
| LCS | G ▷ C72×C17 | |
| UCS | C1 | |
60 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,2,7^2,17|a^2=c,d^a=d^6,e^a=e^6,f^a=f^13,d^b=d^2,e^b=e^4,f^c=f^16104 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊#595), C3×#23756178, C3⋊#23756178 |
| maximal normal subgroup of | ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊#595), C3×#23756178 |
| maximal quotient of | #44472529, C3×#23756178, C3⋊#23756178 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(9996, 81);