Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#62013 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #529×C79 ◁ G | |
1790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 5 generators on 119 points:
(2,5)(3,7)(9,10)(11,15)(12,23)(13,17)(14,19)(16,25)(18,27)(20,34)(21,29)(22,30)(24,32)(26,33)(28,37)(31,38)(35,39)(36,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,28)(17,29)(18,31)(19,30)(23,34)(24,35)(25,37)(26,36)(27,38)(32,39)(33,40);(9,12,14,26)(10,16,19,32)(11,20,22,36)(13,24,27,25)(15,28,30,39)(17,23,18,33)(21,35,38,37)(29,34,31,40);(2,5)(6,8)(9,13,14,27)(10,17,19,18)(11,21,22,38)(12,24,26,25)(15,29,30,31)(16,23,32,33)(20,35,36,37)(28,34,39,40);(1,3,4,7)(2,6,5,8)(11,22)(15,30)(20,36)(21,38)(28,39)(29,31)(34,40)(35,37)3871 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 55001);