Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#60705 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| prime radical | C89×#3455 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×#117×C89 ◁ G | |
1846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 5 generators on 129 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,15,20,28)(12,22,23,37)(13,25,26,17)(16,30,31,24)(19,32,33,40)(21,36,35,29)(27,38,39,34)(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);(2,5)(6,8)(9,11,14,20)(10,15,18,28)(12,19,23,33)(13,21,26,35)(16,27,31,39)(17,29,25,36)(22,32,37,40)(24,34,30,38);(9,12,13,24,14,23,26,30)(10,16,17,22,18,31,25,37)(11,19,21,34,20,33,35,38)(15,27,29,32,28,39,36,40);(1,3,4,7)(2,6,5,8)(11,20)(15,28)(19,33)(21,35)(27,39)(29,36)(32,40)(34,38);(12,23)(16,31)(19,33)(22,37)(24,30)(27,39)(32,40)(34,38)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53693);