Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#60595 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| prime radical | C89×#3010 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C4×#41×C89 ◁ G | |
2014 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,23)(13,16)(14,18)(15,29)(17,26)(19,28)(21,37)(22,32)(24,40)(25,34)(27,36)(30,33)(31,39)(35,38)(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)(9,11,14,13)(10,16,18,20)(12,21,25,24)(15,22,28,27)(17,31,33,35)(19,32,29,36)(23,40,34,37)(26,38,30,39);(9,12,14,25)(10,17,18,33)(11,21,13,24)(15,30,28,26)(16,31,20,35)(19,34,29,23)(22,39,27,38)(32,37,36,40);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,16,18,20)(12,24,25,21)(15,27,28,22)(17,31,33,35)(19,32,29,36)(23,40,34,37)(26,39,30,38);(11,13)(16,20)(21,24)(22,27)(31,35)(32,36)(37,40)(38,39)5429 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53583);