Finite Group DatabasebetaFinite Group Databasebeta| order | 19197 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C79 · Schur multiplier (H2) = C3×C9 |
| direct product | C79×#6879 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C79 | #732 |
| commutator, Frattini, cosocle | C3×C9 | C32×C79 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C79 ◁ C32×C79 ◁ C3×C9×C79 ◁ G | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 106 |
| transitive degree | μt(G) = 2133 |
A minimal faithful permutation representation — 2 generators on 106 points:
(1,2,5)(3,7,12)(4,8,15)(6,16,21)(9,19,23)(10,17,13)(11,20,14)(18,25,27)(22,26,24)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106);(2,6,17,8,18,26,20,19,7)(5,12,23,14,24,25,15,13,16)2765 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19197, 26);