Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#61204 |
| 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 ◁ #532×C89 ◁ G | |
2032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 5 generators on 129 points:
(1,2,4,6)(3,7,8,5)(9,10,13,18)(11,17,21,15)(12,23,24,34)(14,27,26,19)(16,28,32,25)(20,35,36,40)(22,30,38,33)(29,37,31,39)(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)(2,5)(4,8)(6,7)(9,11,13,21)(10,15,18,17)(12,20,24,36)(14,22,26,38)(16,29,32,31)(19,30,27,33)(23,40,34,35)(25,37,28,39);(9,12,13,24)(10,16,18,32)(11,20,21,36)(14,28,26,25)(15,29,17,31)(19,34,27,23)(22,39,38,37)(30,35,33,40);(3,8)(5,7)(9,11)(10,17)(12,20)(13,21)(14,22)(15,18)(16,31)(19,33)(23,35)(24,36)(25,37)(26,38)(27,30)(28,39)(29,32)(34,40);(11,21)(15,17)(20,36)(22,38)(29,31)(30,33)(35,40)(37,39)4895 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 54192);