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×#9766 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | #390 |
| commutator | C42 | C22×C4×C89 |
| Frattini, cosocle | C2×C42 | C23×C89 |
| prime radical | #532×C89 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2,6,9,5,8,4,14)(3,10,13,15,11,16,12,7)(17,18,22,26,23,25,21,19)(20,28,30,32,31,27,29,24)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(1,3)(2,7,8,15)(4,12)(5,11)(6,13)(9,10,14,16)(17,19,22,25,23,26,21,18)(20,28,30,24,31,27,29,32);(2,8)(3,11)(9,14)(12,13)(17,20,22,30,23,31,21,29)(18,24,19,27,25,32,26,28)3560 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 2754);