Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#21737 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | C22×D4 |
| commutator | C2×C4 | C23×C4×C89 |
| Frattini, cosocle | C22×C4 | C24×C89 |
| socle | C22×C89 | C2×C4×D4 |
| prime radical | C23×C89 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C2×C4×C8×C89 ◁ G | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 129 points:
(1,2)(3,8)(4,6)(5,7)(9,10,16,20,14,19,15,31)(11,21,26,36,24,17,25,34)(12,27,30,35,29,39,13,18)(22,37,23,32,28,40,38,33)(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);(1,3,5,4)(2,6,7,8)(9,11,14,24)(10,17,19,21)(12,28,29,22)(13,23,30,38)(15,25,16,26)(18,33,35,32)(20,34,31,36)(27,37,39,40);(9,12,15,13,14,29,16,30)(10,18,20,27,19,35,31,39)(11,22,25,38,24,28,26,23)(17,32,34,40,21,33,36,37);(1,4,5,3)(2,6,7,8)(9,13,15,29,14,30,16,12)(10,18,31,39,19,35,20,27)(11,23,25,22,24,38,26,28)(17,33,36,40,21,32,34,37)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 14725);