Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#61198 |
| 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 ◁ #525×C89 ◁ G | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 5 generators on 129 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,21)(12,25)(13,18)(14,20)(15,19)(16,24)(17,26)(22,36)(23,32)(27,33)(28,31)(29,34)(30,37)(35,39)(38,40)(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)(2,5)(4,8)(6,7)(9,11,14,24)(10,16,20,21)(12,22,27,38)(13,28,15,23)(17,30,34,35)(18,32,19,31)(25,40,33,36)(26,39,29,37);(9,12)(10,17)(11,22)(13,26)(14,27)(15,29)(16,30)(18,33)(19,25)(20,34)(21,35)(23,37)(24,38)(28,39)(31,40)(32,36);(3,8)(5,7)(9,13,14,15)(10,18,20,19)(11,23,24,28)(12,26,27,29)(16,31,21,32)(17,33,34,25)(22,37,38,39)(30,40,35,36);(9,13,14,15)(10,19,20,18)(11,23,24,28)(12,26,27,29)(16,32,21,31)(17,25,34,33)(22,37,38,39)(30,36,35,40)4895 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 54186);