Finite Group DatabasebetaFinite Group Databasebeta| order | 1701 |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C35 |
| unambiguous semidirect product | C7⋊H(5,𝔽3) |
| Fitting | C3×C7×H(3,𝔽3) ⋊ C3 |
| Sylow | C7 ⋊ H(5,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C33×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C3×C7 | C34 |
| Fitting | C3×C7×H(3,𝔽3) | C3 |
| hypercenter | C3×H(3,𝔽3) | ΓL(1,𝔽8) |
| hypocenter | C7 | H(5,𝔽3) |
| LCS | G ▷ C3×C7 ▷ C7 | |
| UCS | C1 ◁ C3 ◁ C3×H(3,𝔽3) | |
586 conjugacy classes of subgroups.
| transitive degree | μt(G) = 189 |
Polycyclic presentation:
3^5,7|b^a=be,d^a=de,c^b=ce,f^d=f^2149 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1701, 304);