Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#11269 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C89 | C2×D4 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| socle | C23×C89 | C2×C4⋊C4 |
| prime radical | C24×C89 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C89 ◁ C22×C42×C89 ◁ G | |
642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 3 generators on 113 points:
(5,6,8,7)(9,10,13,19,15,20,14,18)(11,17,12,16,22,23,21,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(1,2)(3,4)(5,7,8,6)(9,11,14,21,15,22,13,12)(10,16,19,23,20,24,18,17);(2,4)(6,7)(9,12,13,22,15,21,14,11)(10,17,18,24,20,23,19,16)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4257);