Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C22×C4 |
| direct product | C89×#12960 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C89 |
| Frattini, cosocle | C22×D4 | C23×C89 |
| prime radical | C2×#120×C89 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C89 ◁ C22×D4×C89 ◁ G | |
608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2,4,3)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(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);(1,3,4,2)(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19);(7,14)(8,16)(10,17)(13,19)3026 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 5948);