Finite Group DatabasebetaFinite Group Databasebeta| order | 34356 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊#13109888 |
| Fitting, socle, derived, Hall | C7×C409 ⋊ C3×C4 |
| Hall | C7⋊#13109888 ⋊ C3 |
| Hall | ΓL(1,𝔽8)×C409 ⋊ C4 |
| Sylow | C409 ⋊ ΓL(1,𝔽8)⋊C4 |
| Hall | ΓL(1,𝔽8) ⋊ #13109888 |
| Sylow | C7 ⋊ C3×#13109888 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C7×C409 | C3×C4 |
| prime radical | ΓL(1,𝔽8)×D409 | C2 |
| cosocle | C7×D409 | C2×C3 |
| LCS | G ▷ C7×C409 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2863 |
Polycyclic presentation:
2,3,2,7,409|a^2=c,d^a=d^6,e^a=e^266,d^b=d^2,e^c=e^408524 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34356, 31);