Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#23464 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C24×C79 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C23×C4×C79 ◁ G | |
1146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,24)(13,17)(14,19)(16,25)(18,28)(20,36)(21,30)(22,35)(23,31)(26,33)(27,34)(29,38)(32,37)(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);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,20)(13,21)(14,23)(16,32)(17,30)(18,35)(19,31)(22,28)(24,36)(25,37)(26,39)(27,38)(29,34)(33,40);(9,12,14,26)(10,16,19,34)(11,20,23,39)(13,27,28,25)(15,29,31,32)(17,24,18,33)(21,38,22,37)(30,40,35,36);(9,13,14,28)(10,17,19,18)(11,21,23,22)(12,25,26,27)(15,30,31,35)(16,33,34,24)(20,37,39,38)(29,36,32,40)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16452);