Finite Group DatabasebetaFinite Group Databasebeta| order | 1647 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C61 · Schur multiplier (H2) = C32 |
| direct product | H(3,𝔽3)×C61 |
| unambiguous semidirect product | C32⋊(C3×C61) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C61 | C32 |
| commutator, Frattini, cosocle | C3 | C32×C61 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C61 ◁ G | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 549 |
A minimal faithful permutation representation — 2 generators on 70 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70);(2,5,6)(4,8,9)671 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1647, 3);