Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C2×C42 |
| direct product | C89×#13231 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | #1964 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C2×#41 | C23×C89 |
| prime radical | C89×#3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C2×C4×C89 ◁ C2×#131×C89 ◁ G | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2)(3,5,13,20)(4,14)(6,21)(7,10)(8,17)(9,15)(11,23,26,32)(12,27,24,19)(16,30,31,25)(18,28)(22,29)(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);(1,3,5,10,7,13,20,2)(4,11,16,22,18,26,31,8)(6,12,19,21,15,24,27,9)(14,17,25,23,28,29,30,32);(1,4,15,29,7,18,6,17)(2,8,21,28,10,22,9,14)(3,11,24,30,13,26,12,25)(5,16,27,32,20,31,19,23)3026 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 6219);