Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#62036 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ #529×C89 ◁ G | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 5 generators on 129 points:
(2,5)(3,7)(9,10,15,20)(11,16,23,31)(12,25,27,34)(13,19,14,18)(17,28,33,26)(21,35,37,40)(22,32,24,30)(29,38,39,36)(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);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,22)(14,24)(15,23)(17,29)(18,30)(19,32)(20,31)(25,35)(26,36)(27,37)(28,38)(33,39)(34,40);(9,12,15,27)(10,17,20,33)(11,21,23,37)(13,26,14,28)(16,29,31,39)(18,25,19,34)(22,36,24,38)(30,35,32,40);(2,5)(6,8)(9,13,15,14)(10,18,20,19)(11,22,23,24)(12,26,27,28)(16,30,31,32)(17,25,33,34)(21,36,37,38)(29,35,39,40);(1,3,4,7)(2,6,5,8)(11,23)(16,31)(21,37)(22,24)(29,39)(30,32)(35,40)(36,38)4361 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 55024);