Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C26 |
| direct product | C89×#23531 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | C22×AGL(1,ℤ/8)×C89 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C4×C89 ◁ G | |
1454 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)(7,15)(8,11)(9,13)(10,14)(12,18)(16,20)(17,19)(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);(6,11)(7,16)(8,9)(12,15)(13,14)(19,20);(2,4)(5,7,8,17,10,16,9,18)(6,12,13,20,14,19,11,15);(7,16)(12,19)(15,20)(17,18)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 16519);