Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#25050 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | C89×#3060 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C22×C4×C89 ◁ G | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 129 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,21)(12,25)(13,29)(14,20)(15,19)(16,23)(17,24)(18,26)(22,36)(27,39)(28,34)(30,32)(31,38)(33,40)(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);(1,3,4,7)(2,5,6,8)(9,11,15,30,14,24,13,23)(10,16,29,17,20,32,19,21)(12,22,28,40,27,38,26,37)(18,31,39,33,34,36,25,35);(2,6)(5,8)(9,11)(10,17)(12,22)(13,23)(14,24)(15,30)(16,19)(18,33)(20,21)(25,31)(26,37)(27,38)(28,40)(29,32)(34,35)(36,39);(9,12,13,26,14,27,15,28)(10,18,19,25,20,34,29,39)(11,22,23,37,24,38,30,40)(16,31,21,35,32,36,17,33)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 18038);