Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C2 |
| direct product | C89×#8804 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | #383 |
| commutator | C42 | C22×C4×C89 |
| Frattini, cosocle | C2×C42 | C23×C89 |
| prime radical | C23×C89 | #126 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
216 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)(3,10,12,14)(4,6)(5,9)(7,11,15,16)(8,13)(17,18)(19,27,29,32)(20,24,30,23)(21,25)(22,26)(28,31)(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);(2,6,9,8)(7,10,15,14)(17,19,28,30,22,29,21,20)(18,23,25,27,26,24,31,32);(1,3,4,11,5,12,13,16)(2,7,8,14,9,15,6,10)(17,20,21,29,22,30,28,19)(18,24,25,32,26,23,31,27)3560 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 1792);