Finite Group DatabasebetaFinite Group Databasebeta| order | 19197 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C79 · Schur multiplier (H2) = C3 |
| direct product | C79×#6899 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C79 | C33 |
| commutator, Frattini, cosocle | C32 | C33×C79 |
| prime radical | C33×C79 | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C79 ◁ G | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 6399 |
A minimal faithful permutation representation — 3 generators on 115 points:
(2,5,3)(6,7,8)(10,11,15)(12,19,31)(13,17,25)(14,18,26)(16,27,22)(20,30,34)(21,32,24)(23,29,35)(28,36,33)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,2,6,9,3,7,4,5,8)(10,12,20,14,21,33,23,22,13)(11,16,28,18,19,17,29,32,30)(15,24,25,26,27,34,35,31,36);(1,3,8,9,5,6,4,2,7)(10,13,22,23,33,21,14,20,12)(11,17,16,29,28,32,18,30,19)(15,25,27,35,36,24,26,34,31)4029 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19197, 46);