Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C79×#24525 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C23×C4×C79 ◁ G | |
858 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)(4,6)(5,8)(9,10,15,29,14,19,13,18)(11,16,22,32,23,33,21,31)(12,24,26,17,28,35,27,36)(20,37,25,30,39,34,38,40)(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);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,25)(13,21)(14,23)(15,22)(17,34)(18,31)(19,33)(20,27)(24,30)(26,39)(28,38)(29,32)(35,40)(36,37);(9,12)(10,17)(11,20)(13,26)(14,28)(15,27)(16,30)(18,35)(19,36)(21,25)(22,38)(23,39)(24,29)(31,34)(32,37)(33,40);(2,6)(5,7)(9,13,14,15)(10,18,19,29)(11,21,23,22)(12,26,28,27)(16,31,33,32)(17,35,36,24)(20,25,39,38)(30,34,40,37)3397 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 17513);