Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#26851 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C2×AGL(1,ℤ/8)×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ G | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,23)(12,27)(13,32)(14,21)(15,20)(16,22)(17,24)(18,30)(19,33)(25,37)(26,36)(28,38)(29,39)(31,35)(34,40)(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);(9,11,15,25,16,26,14,24)(10,17,21,36,22,37,20,23)(12,28,30,34,31,33,29,13)(18,38,27,32,39,19,35,40);(2,5)(6,8)(9,12,14,29,16,31,15,30)(10,18,20,35,22,39,21,27)(11,13,24,33,26,34,25,28)(17,32,23,38,37,40,36,19);(1,3,4,7)(2,6,5,8)(9,13,14,33,16,34,15,28)(10,19,20,32,22,38,21,40)(11,12,24,29,26,31,25,30)(17,35,23,39,37,27,36,18)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 19839);