Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#61492 |
| 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 | |
1770 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)(6,8)(9,10)(11,15)(12,23)(13,17)(14,19)(16,26)(18,28)(20,36)(21,30)(22,31)(24,40)(25,34)(27,35)(29,38)(32,39)(33,37)(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);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,24)(13,21)(14,22)(16,33)(17,30)(18,32)(19,31)(20,27)(23,40)(25,38)(26,37)(28,39)(29,34)(35,36);(9,12)(10,16)(11,20)(13,25)(14,27)(15,29)(17,23)(18,35)(19,34)(21,37)(22,24)(26,28)(30,36)(31,33)(32,40)(38,39);(2,5)(6,8)(9,13,14,28)(10,17,19,18)(11,21,22,39)(12,25,27,26)(15,30,31,32)(16,23,34,35)(20,37,24,38)(29,36,33,40);(1,3,4,7)(2,6,5,8)(11,22)(15,31)(20,24)(21,39)(29,33)(30,32)(36,40)(37,38)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54480);