Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C2×C4 |
| direct product | C89×#9751 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | #121 |
| commutator | C42 | C22×C4×C89 |
| Frattini, cosocle | C2×C42 | C23×C89 |
| socle | C22×C89 | #390 |
| prime radical | #532×C89 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C2×C42×C89 ◁ G | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 113 points:
(1,2,5,6,4,7,3,8)(9,10,13,17,14,18,12,15)(11,19,21,24,22,23,20,16)(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);(2,6)(3,5)(7,8)(10,15,18,17)(16,23,24,19);(2,7)(6,8)(9,11,12,20,14,22,13,21)(10,16,17,19,18,24,15,23)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 2739);