Finite Group DatabasebetaFinite Group Databasebeta| order | 19197 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C79 · Schur multiplier (H2) = C32 |
| direct product | C9×3−1+2×C79 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C79 | C32 |
| commutator | C3 | C32×C9×C79 |
| Frattini, cosocle | C32 | C33×C79 |
| socle | C32×C79 | C33 |
| prime radical | C33×C79 | C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C79 ◁ G | |
148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 6399 |
A minimal faithful permutation representation — 3 generators on 97 points:
(1,2,4,3,5,7,6,8,9)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(10,11,13,12,15,17,16,14,18);(11,14,15)(13,17,18)7821 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19197, 36);