Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#33681 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C89 |
| prime radical | #466×C89 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ #117×C89 ◁ #405×C89 ◁ G | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 129 points:
(1,2)(3,6)(4,5)(7,8)(11,19)(12,22)(13,24)(15,27)(17,32)(18,31)(20,37)(21,36)(23,26)(25,35)(28,39)(29,40)(30,38)(33,34)(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)(9,10)(11,15)(12,17)(13,18)(14,16)(19,27)(20,29)(21,28)(22,32)(23,33)(24,31)(25,30)(26,34)(35,38)(36,39)(37,40);(9,11,14,21)(10,15,16,28)(12,23,25,37)(13,26,24,20)(17,33,30,40)(18,34,31,29)(19,35,36,22)(27,38,39,32);(1,3,4,7)(2,6,5,8)(10,16)(15,28)(17,30)(18,31)(27,39)(29,34)(32,38)(33,40)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26669);