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×#26271 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ G | |
752 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)(3,7)(4,5)(6,8)(9,10)(11,22)(12,26)(13,31)(14,33)(15,20)(16,21)(17,24)(18,29)(19,27)(23,40)(25,36)(28,37)(30,35)(32,38)(34,39)(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);(2,5)(6,7)(9,11,14,24,16,25,15,28)(10,17,20,22,21,37,33,36)(12,27,29,23,30,32,34,13)(18,38,26,40,39,19,35,31);(9,12,14,29,16,30,15,34)(10,18,20,26,21,39,33,35)(11,23,24,32,25,13,28,27)(17,31,22,38,37,40,36,19);(1,3,4,8)(2,6,5,7)(9,13,15,32,16,23,14,27)(10,19,33,40,21,38,20,31)(11,12,28,34,25,30,24,29)(17,35,36,39,37,26,22,18)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 19259);