Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C22×C4 |
| direct product | C89×#29138 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C89 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ #376×C89 ◁ G | |
642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 22784 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,22,25,21,24,29,32)(19,20,27,28,26,23,31,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,32)(29,31);(11,15)(13,16)(17,20,21,23)(18,19,24,26)(22,30,29,28)(25,31,32,27);(1,3,4,8)(2,6,5,7)(18,24)(19,26)(25,32)(27,31)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 22126);