Finite Group DatabasebetaFinite Group Databasebeta| order | 1029 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C72 · Schur multiplier (H2) = C1 |
| direct product | C3×7−1+2 |
| unambiguous semidirect product | (C3×C49)⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C7 | C72 |
| commutator, Frattini, cosocle | C7 | C3×C72 |
| prime radical | C3×C72 | C7 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C3×C7 ◁ G | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 147 |
A minimal faithful permutation representation — 2 generators on 52 points:
(1,2,3)(4,5,7,11,18,29,38,6,9,13,20,31,40,47,10,16,23,33,42,39,48,17,27,22,35,30,41,49,28,26,25,19,32,43,50,37,15,12,21,34,44,51,46,8,14,24,36,45,52);(5,8,15,26,27,16,9)(7,12,22,13,14,25,23)(11,19,20,21,33,24,35)(18,30,36,42,34,31,32)(29,39,43,45,40,41,44)(38,47,48,49,50,51,52)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1029, 4);