Finite Group DatabasebetaFinite Group Databasebeta| order | 2133 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C79 · Schur multiplier (H2) = C1 |
| direct product | 3−1+2×C79 |
| unambiguous semidirect product | C9⋊(C3×C79) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C79 | C32 |
| commutator, Frattini, cosocle | C3 | C32×C79 |
| prime radical | C32×C79 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C79 ◁ G | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 88 |
| transitive degree | μt(G) = 711 |
A minimal faithful permutation representation — 2 generators on 88 points:
(1,2,4,3,6,8,7,5,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88);(2,5,6)(4,8,9)869 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2133, 4);