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×#6893 |
| 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 | |
156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 567 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(8,9,11)(10,13,15)(12,16,14)(17,18,21)(19,23,29)(20,24,28)(22,30,27)(25,32,37)(26,33,36)(31,38,35)(34,40,42)(39,43,41);(18,22,24)(21,28,30)(23,31,33)(29,36,38)(32,39,40)(37,42,43);(9,12,13)(11,15,16)(17,19,25,20,26,34,27,35,41)(18,23,32,24,33,40,22,31,39)(21,29,37,28,36,42,30,38,43)357 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1701, 40);