Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C89 · Schur multiplier (H2) = C29 |
| direct product | C89×#60146 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | C25 |
| commutator | C22 | C24×C4×C89 |
| Frattini, cosocle | C2×C4 | C25×C89 |
| socle | C22×C89 | C24×C4 |
| prime radical | D4×#41×C89 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ G | |
2756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 5 generators on 109 points:
(1,2)(3,4)(5,6,9,7,10,12,17,14)(8,11,16,13,15,18,20,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(2,4);(5,7,9,12,10,14,17,6)(8,13,16,18,15,19,20,11);(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20);(8,15)(11,18)(13,19)(16,20)7832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53134);