Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C89 · Schur multiplier (H2) = C2×C4 |
| direct product | C89×#11498 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C89 |
| Frattini, cosocle | C22×C8 | C23×C89 |
| socle | C22×C89 | C2×C22⋊C8 |
| prime radical | C23×C89 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C2×C4×C8×C89 ◁ G | |
292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 3 generators on 137 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,28,24,27,37,45,23,29,36,46,21,38,20,26)(19,30,33,44,32,25,31,41,34,43,47,40,39,42,35,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,34)(18,25,29,42)(20,35,37,31)(21,39,24,32)(22,33,36,47)(26,44,45,40)(27,43,38,30)(28,41,46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,21,36,23,37,24,22)(18,26,38,46,29,45,27,28)(19,31,39,33,34,35,32,47)(25,40,30,41,42,44,43,48)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4486);