Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#23658 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C89 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | C24×C89 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C22×C42×C89 ◁ G | |
990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 109 points:
(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(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);(5,7,9,8)(6,10,11,12)(13,14)(15,19)(16,18)(17,20);(2,4)(5,8,9,7)(6,10,11,12)(14,18)(15,16)(19,20);(13,15,17,16)(14,19,20,18)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 16646);