Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C27 |
| direct product | C89×#34629 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C89 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C89 |
| socle | C23×C89 | C25 |
| prime radical | C24×C89 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C89 ◁ G | |
1558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 5 generators on 121 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,27)(21,23)(24,28)(25,30)(26,32)(29,31)(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)(6,8)(9,11,12,15)(10,13,14,16)(17,19,21,25)(18,22,23,30)(20,26,28,31)(24,29,27,32);(10,14)(13,16)(18,23)(19,25)(24,27)(26,31);(17,20,21,28)(18,24,23,27)(19,26,25,31)(22,29,30,32);(1,3,4,7)(2,6,5,8)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 27617);