Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#18827 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | C79×#3499 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,15)(12,22)(13,25)(14,17)(16,24)(18,28)(19,35)(20,33)(21,30)(23,34)(26,31)(27,32)(29,37)(36,39)(38,40)(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);(2,5)(3,8)(6,7)(9,11,14,21)(10,15,17,30)(12,19,23,36)(13,26,28,20)(16,29,32,40)(18,33,25,31)(22,39,34,35)(24,38,27,37);(9,12)(10,16)(11,19)(13,27)(14,23)(15,29)(17,32)(18,34)(20,38)(21,36)(22,25)(24,28)(26,37)(30,40)(31,39)(33,35);(1,3,4,8)(2,5,7,6)(10,17)(15,30)(16,32)(18,25)(22,34)(29,40)(31,33)(35,39)2923 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11815);