Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C22 |
| direct product | C89×#13128 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | #117 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C22×C8 | C23×C89 |
| socle | C22×C89 | #432 |
| prime radical | C24×C89 | M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C2×#97×C89 ◁ G | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2)(3,4)(5,6,9,14,8,13,16,20,10,7,11,15,17,12,18,19)(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);(5,7,9,14,8,12,16,20,10,6,11,15,17,13,18,19);(2,4)(6,12,7,13)(9,11)(14,20,15,19)(16,18)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 6116);