Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C2×C4 |
| direct product | C89×#9755 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | #390 |
| commutator | C42 | C22×C4×C89 |
| Frattini, cosocle | C2×C42 | C23×C89 |
| prime radical | C23×C89 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 129 points:
(2,5)(3,7)(6,8)(9,10)(11,15,20,29)(12,22)(13,25)(14,17)(16,23)(18,26)(19,31,32,40)(21,35,34,28)(24,36)(27,38,39,33)(30,37)(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);(1,2,3,5,4,6,7,8)(9,11)(10,15,17,29)(12,19)(13,21)(14,20)(16,27,30,39)(18,28,25,35)(22,31,36,40)(23,33)(24,32)(26,34)(37,38);(9,12,13,23,14,24,26,37)(10,16,18,22,17,30,25,36)(11,19,21,33,20,32,34,38)(15,27,28,31,29,39,35,40)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 2743);