Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#19013 |
| 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 | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 127 points:
(2,6)(4,11)(7,13)(8,12)(9,15)(14,16)(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);(4,11)(6,12)(9,15)(13,16)(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);(1,2)(3,7)(4,12)(5,8)(6,11)(9,16)(10,14)(13,15)(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);(1,3,5,10)(2,7,8,14)(4,9,11,15)(6,13,12,16)(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, 12001);