Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27×C89 · Schur multiplier (H2) = C220 |
| direct product | C22×#585×C89 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C89 | C24 |
| commutator, Frattini, cosocle | C2 | C27×C89 |
| socle | C23×C89 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C89 ◁ G | |
60366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 7 generators on 109 points:
(1,2)(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);(3,4);(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19);(7,14)(10,13)(15,20)(17,19);(6,11)(10,13)(12,18)(17,19);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)12104 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 56088);