Finite Group DatabasebetaFinite Group Databasebeta| order | 19197 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C79 · Schur multiplier (H2) = C32 |
| direct product | C3×C79×C27⋊C3 |
| unambiguous semidirect product | C27⋊(C32×C79) |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C79 | C32 |
| commutator | C3 | C32×C9×C79 |
| Frattini, cosocle | C9 | C33×C79 |
| socle | C32×C79 | C3×C9 |
| prime radical | C33×C79 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C79 ◁ G | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 6399 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2,3)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(4,5,8,6,10,16,12,19,24,7,11,15,13,20,23,21,27,29,14,9,17,22,18,25,28,26,30);(5,9,11)(8,15,17)(10,18,20)(16,23,25)(19,26,27)(24,29,30)7821 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19197, 49);