Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C27 |
| direct product | C79×#36700 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C79 |
| socle | C23×C79 | C25 |
| prime radical | C24×C79 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ G | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 5 generators on 119 points:
(1,2)(3,6)(4,8)(5,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,29,31)(27,33,34,30)(28,32,36,38)(35,39,40,37)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(1,3,5,4)(2,6,7,8)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(25,27,29,34)(26,30,31,33)(28,35,36,40)(32,37,38,39);(2,7)(6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(27,34)(30,33)(35,40)(37,39);(1,4,5,3)(2,8,7,6)(10,16)(12,20)(14,22)(17,19)(26,31)(30,33)(32,38)(37,39);(25,28,29,36)(26,32,31,38)(27,35,34,40)(30,37,33,39)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 29688);