Finite Group DatabasebetaFinite Group Databasebeta| order | 19197 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C79 · Schur multiplier (H2) = C3 |
| direct product | C79×#6861 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C79 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C32×C79 |
| prime radical | C33×C79 | C32 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C79 ◁ C33×C79 ◁ G | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 6399 |
A minimal faithful permutation representation — 2 generators on 133 points:
(1,2,5)(3,9,13)(4,8,14)(6,15,11)(7,18,12)(10,20,23)(16,26,27)(17,25,22)(19,24,21)(28,29,32,31,34,41,39,45,51)(30,35,46,38,33,42,37,44,40)(36,43,50,48,53,54,49,47,52)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133);(1,3,10,4,11,21,12,22,27)(2,6,16,8,17,20,7,9,19)(5,13,23,14,15,24,18,25,26)(28,30,36)(29,33,43)(31,38,48)(32,40,50)(34,44,53)(35,47,45)(37,49,39)(41,46,54)(42,52,51)2765 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19197, 8);