Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#26651 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C2×#41×C79 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ G | |
766 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)(9,10,13,19,14,20,15,21)(11,22,25,33,26,34,27,16)(12,28,29,36,30,17,31,18)(23,37,24,32,38,35,39,40)(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);(9,11,14,26)(10,16,20,33)(12,24,30,39)(13,27,15,25)(17,35,28,37)(18,32,36,40)(19,34,21,22)(23,31,38,29);(2,5)(6,8)(9,12)(10,17)(11,23)(13,29)(14,30)(15,31)(16,32)(18,19)(20,28)(21,36)(22,35)(24,25)(26,38)(27,39)(33,40)(34,37);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,24)(13,31)(14,30)(15,29)(16,32)(17,19)(20,36)(21,28)(22,37)(23,25)(26,39)(27,38)(33,40)(34,35)3397 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 19639);