Finite Group DatabasebetaFinite Group Databasebeta| order | 1701 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C7 · Schur multiplier (H2) = C33 |
| direct product | C7×C9⋊H(3,𝔽3) |
| unambiguous semidirect product | C9⋊(C7×H(3,𝔽3)) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C7 | C33 |
| commutator, Frattini, cosocle | C32 | C33×C7 |
| prime radical | C3×C7×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C7 ◁ G | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 567 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5,6,7)(9,11,14)(12,16,15)(17,18,20)(19,22,24)(21,25,23);(8,9,12,10,11,15,13,14,16);(18,21,22)(20,24,25)357 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1701, 39);