Finite Group DatabasebetaFinite Group Databasebeta| order | 34524 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(ΓL(1,𝔽8)⋊#10587739) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C3⋊#10587739) |
| unambiguous semidirect product | (C3⋊(C7⋊#10587739))⋊C3 |
| Fitting, socle, derived | C3×C7×C137 ⋊ C3×C4 |
| Hall | C3×ΓL(1,𝔽8)×C137 ⋊ C4 |
| Hall | C7×C137 ⋊ C3×Dic3 |
| Sylow | C137 ⋊ C3⋊(ΓL(1,𝔽8)⋊C4) |
| Hall | C3×ΓL(1,𝔽8) ⋊ #10587739 |
| Sylow | C7 ⋊ C3×C3⋊#10587739 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C7×C137 | C3×C4 |
| prime radical | C3×ΓL(1,𝔽8)×D137 | C2 |
| cosocle | C3×C7×D137 | C2×C3 |
| LCS | G ▷ C3×C7×C137 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2877 |
Polycyclic presentation:
2,3,2,3,7,137|a^2=c,d^a=d^2,e^a=e^6,f^a=f^100,e^b=e^2,f^c=f^136534 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34524, 65);