Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#18994 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | C79×#3153 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 119 points:
(2,5)(3,7)(6,8)(9,10)(11,19)(12,16)(13,25)(14,17)(15,21)(18,28)(20,36)(22,33)(23,34)(24,31)(26,30)(27,32)(29,37)(35,38)(39,40)(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);(3,7)(5,8)(9,11)(10,15)(12,20)(13,26)(14,22)(16,29)(17,30)(18,33)(19,25)(21,28)(23,39)(24,38)(27,37)(31,40)(32,35)(34,36);(1,2)(3,8)(4,6)(5,7)(9,12,14,24)(10,16,17,31)(11,20,22,38)(13,27,28,23)(15,29,30,40)(18,34,25,32)(19,35,33,36)(21,39,26,37);(10,17)(15,30)(16,31)(18,25)(19,33)(29,40)(32,34)(35,36)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11982);