Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#19005 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | C79×#3531 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 127 points:
(3,9)(4,11)(6,14)(7,12)(10,13)(15,16)(17,18,22,26)(19,27,31,38)(20,24,34,42)(21,35,36,25)(23,29,39,30)(28,44,46,43)(32,41,33,40)(37,47,48,45)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,2,5,8)(3,6,10,15)(4,12,11,7)(9,16,13,14)(17,19,22,31)(18,23,26,39)(20,28,34,46)(21,29,36,30)(24,37,42,48)(25,38,35,27)(32,45,33,47)(40,44,41,43);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,11)(12,14)(17,20)(18,24)(19,28)(21,32)(22,34)(23,37)(25,40)(26,42)(27,43)(29,45)(30,47)(31,46)(33,36)(35,41)(38,44)(39,48);(17,21,22,36)(18,25,26,35)(19,29,31,30)(20,32,34,33)(23,38,39,27)(24,40,42,41)(28,45,46,47)(37,44,48,43)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11993);