Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#33878 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C79×#3280 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C4×D4×C79 ◁ C8×D4×C79 ◁ G | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,23)(13,27)(14,19)(16,25)(17,21)(18,22)(20,34)(24,38)(26,39)(28,32)(29,31)(30,36)(33,40)(35,37)(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)(2,5)(4,8)(6,7)(9,11,14,22)(10,15,19,18)(12,24,26,20)(13,21,29,37)(16,32,33,30)(17,31,35,27)(23,38,39,34)(25,28,40,36);(9,12)(10,16)(11,20)(13,28)(14,26)(15,30)(17,34)(18,32)(19,33)(21,25)(22,24)(23,27)(29,36)(31,39)(35,38)(37,40);(3,8)(5,7)(9,11,14,22)(10,15,19,18)(12,20,26,24)(13,21,29,37)(16,30,33,32)(17,31,35,27)(23,34,39,38)(25,36,40,28)3397 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26866);