Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C27×C4 |
| direct product | C89×#60823 |
| 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 ◁ C4×#41×C89 ◁ G | |
2054 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)(3,7)(4,6)(5,8)(9,10)(11,21)(12,27)(13,19)(14,20)(15,31)(16,25)(17,28)(18,29)(22,37)(23,33)(24,38)(26,34)(30,36)(32,40)(35,39)(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);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,23)(13,24)(14,25)(15,26)(17,35)(18,33)(19,34)(20,21)(22,29)(27,37)(28,40)(30,39)(31,38)(32,36);(3,8)(5,7)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,32)(18,19)(20,36)(21,35)(23,25)(24,39)(26,40)(27,31)(33,38)(34,37);(2,6)(5,7)(9,12)(10,18)(11,23)(13,30)(14,29)(15,28)(16,33)(17,19)(20,27)(21,37)(22,25)(24,39)(26,40)(31,36)(32,38)(34,35);(9,13,14,15)(10,19,20,31)(11,24,25,26)(12,28,29,30)(16,34,21,38)(17,18,36,27)(22,39,23,40)(32,37,35,33)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53811);