Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#60499 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| prime radical | C2×2−1+4×C89 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C4×#41×C89 ◁ G | |
1970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 5 generators on 129 points:
(1,2,4,6)(3,7,8,5)(9,10,13,15)(11,19,21,32)(12,17,23,29)(14,26,25,18)(16,27,28,22)(20,33,34,40)(24,37,36,31)(30,38,39,35)(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)(10,15)(16,28)(17,29)(18,26)(19,32)(30,39)(31,37)(33,40);(9,11,13,21)(10,16,15,28)(12,20,23,34)(14,27,25,22)(17,30,29,39)(18,32,26,19)(24,38,36,35)(31,40,37,33);(9,12)(10,17)(11,20)(13,23)(14,24)(15,29)(16,30)(18,31)(19,33)(21,34)(22,35)(25,36)(26,37)(27,38)(28,39)(32,40);(12,23)(17,29)(20,34)(24,36)(30,39)(31,37)(33,40)(35,38)5429 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53487);