Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#21165 |
| 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 | C24×C89 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C22×M16×C89 ◁ G | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 4 generators on 109 points:
(1,2)(3,4)(5,6,10,14,11,15,9,19)(7,12,17,20,16,13,8,18)(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)(5,7,10,17,11,16,9,8)(6,12,14,20,15,13,19,18);(5,8,9,16,11,17,10,7)(6,13,14,18,15,12,19,20);(7,16)(8,17)(12,13)(18,20)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 14153);