Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#19097 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | C79×#3567 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 127 points:
(3,10)(4,13)(6,14)(7,8)(11,12)(15,16)(17,18,22,26)(19,27,30,42)(20,24,33,41)(21,34,36,25)(23,35,38,29)(28,44,47,43)(31,40,32,39)(37,45,48,46)(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)(3,6)(4,7)(5,9)(8,13)(10,16)(11,14)(12,15)(17,19,22,30)(18,23,26,38)(20,28,33,47)(21,35,36,29)(24,37,41,48)(25,42,34,27)(31,46,32,45)(39,44,40,43);(1,3)(2,6)(4,11)(5,12)(7,14)(8,16)(9,15)(10,13)(17,20)(18,24)(19,28)(21,31)(22,33)(23,37)(25,39)(26,41)(27,43)(29,45)(30,47)(32,36)(34,40)(35,46)(38,48)(42,44);(1,4,5,13)(2,7,9,8)(3,11,12,10)(6,14,15,16)(17,21,22,36)(18,25,26,34)(19,29,30,35)(20,31,33,32)(23,27,38,42)(24,39,41,40)(28,45,47,46)(37,43,48,44)2449 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 12085);