Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#62048 |
| 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 | |
1796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 5 generators on 129 points:
(2,5)(3,7)(9,10,14,19)(11,21,24,16)(12,25,27,34)(13,20,15,18)(17,29,32,26)(22,36,38,40)(23,31,28,33)(30,39,35,37)(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)(12,22)(13,28)(14,24)(15,23)(17,30)(18,33)(19,21)(20,31)(25,40)(26,39)(27,38)(29,37)(32,35)(34,36);(9,12)(10,17)(11,22)(13,26)(14,27)(15,29)(16,30)(18,25)(19,32)(20,34)(21,35)(23,37)(24,38)(28,39)(31,36)(33,40);(2,5)(6,8)(9,13,14,15)(10,18,19,20)(11,23,24,28)(12,26,27,29)(16,31,21,33)(17,25,32,34)(22,37,38,39)(30,36,35,40);(1,3,4,7)(2,6,5,8)(10,19)(11,24)(17,32)(18,20)(22,38)(23,28)(25,34)(37,39)4361 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 55036);