Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#18972 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | #121×C79 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
818 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)(10,15)(14,16)(17,18,22,26)(19,23,30,40)(20,31,33,41)(21,35,36,25)(24,32,42,34)(27,43,45,47)(28,39,29,38)(37,46,48,44)(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)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,19)(18,23)(20,27)(21,28)(22,30)(24,37)(25,38)(26,40)(29,36)(31,47)(32,44)(33,45)(34,46)(35,39)(41,43)(42,48);(1,3,5,9)(2,7,8,14)(4,10,11,15)(6,13,12,16)(17,20)(18,24)(19,27)(21,32)(22,33)(23,37)(25,41)(26,42)(28,44)(29,46)(30,45)(31,35)(34,36)(38,43)(39,47)(40,48);(3,9)(7,14)(10,15)(13,16)(17,21,22,36)(18,25,26,35)(19,28,30,29)(20,32,33,34)(23,38,40,39)(24,41,42,31)(27,44,45,46)(37,43,48,47)2923 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11960);