Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#33677 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C89 |
| prime radical | #466×C89 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ C2×C4×C8×C89 ◁ G | |
548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 129 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,15)(13,25)(14,28)(16,21)(17,27)(18,24)(20,35)(22,39)(23,36)(26,34)(29,33)(30,37)(31,38)(32,40)(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)(10,15)(16,30)(17,31)(18,28)(19,35)(25,36)(32,34)(33,39);(9,11,12,20)(10,16,15,30)(13,26,23,40)(14,29,24,22)(17,33,31,39)(18,34,28,32)(19,36,35,25)(21,38,37,27);(1,3,4,7)(2,6,5,8)(11,20)(16,30)(19,35)(21,37)(22,29)(26,40)(32,34)(33,39)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26665);