Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#62072 |
| 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 | |
1780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 5 generators on 119 points:
(2,5)(3,7)(9,10,14,18)(11,20,23,15)(12,24,26,33)(13,17,28,19)(16,25,32,27)(21,35,37,40)(22,31,39,30)(29,38,34,36)(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)(12,21)(13,22)(14,23)(16,29)(17,30)(18,20)(19,31)(24,40)(25,36)(26,37)(27,38)(28,39)(32,34)(33,35);(9,12,14,26)(10,16,18,32)(11,21,23,37)(13,25,28,27)(15,29,20,34)(17,24,19,33)(22,36,39,38)(30,40,31,35);(2,5)(6,8)(9,13,14,28)(10,17,18,19)(11,22,23,39)(12,25,26,27)(15,30,20,31)(16,24,32,33)(21,36,37,38)(29,40,34,35);(1,3,4,7)(2,6,5,8)(10,18)(11,23)(16,32)(17,19)(21,37)(22,39)(24,33)(36,38)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 55060);