Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C89 · Schur multiplier (H2) = C2×C4 |
| direct product | C89×#7530 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | #1408 |
| commutator | C4×C8 | C2×C4×C89 |
| Frattini, cosocle | #493 | C22×C89 |
| prime radical | #493×C89 | C22 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C22×C89 ◁ C2×C4×C89 ◁ C42×C89 ◁ #493×C89 ◁ G | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 2 generators on 121 points:
(1,2)(3,11,13,28)(4,15,18,25)(5,10)(6,9,17,27)(7,12,21,31)(8,19,24,30)(14,29,20,23)(16,26)(22,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(1,3,5,14)(2,7,10,21)(4,16,19,32)(6,20,17,13)(8,23,25,28)(9,26,27,22)(11,15,29,24)(12,30,31,18)2225 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 518);