Finite Group DatabasebetaFinite Group Databasebeta| order | 19197 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C79 · Schur multiplier (H2) = C33 |
| direct product | C3×C79×#733 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C79 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C33×C79 |
| prime radical | C33×C79 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C79 ◁ C33×C79 ◁ G | |
132 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);(4,5,8,7,11,17,15,10,21)(6,12,16,14,9,18,25,20,28)(13,23,26,24,22,27,30,19,29);(4,6,13,15,25,30,7,14,24)(5,9,19,10,12,22,11,20,23)(8,16,26,21,28,29,17,18,27)4029 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19197, 54);