Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#25529 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | #525×C89 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C4×C89 ◁ G | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 129 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,23)(13,27)(14,18)(15,19)(17,26)(20,34)(21,32)(22,33)(24,39)(25,36)(28,31)(29,35)(30,38)(37,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(1,3,4,8)(2,5,6,7)(9,11,15,22)(10,16,19,33)(12,24,25,38)(13,21,14,28)(17,34,35,40)(18,31,27,32)(20,29,37,26)(23,39,36,30);(9,12,13,26,15,25,14,29)(10,17,18,23,19,35,27,36)(11,20,21,38,22,37,28,24)(16,30,31,34,33,39,32,40);(2,6)(5,7)(12,25)(17,35)(20,37)(23,36)(24,38)(26,29)(30,39)(34,40)3560 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 18517);