Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#33618 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | C2×#466 |
| commutator, Frattini, cosocle | C2×C8 | C24×C89 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C22×C4×C89 ◁ C2×#131×C89 ◁ G | |
656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 4 generators on 121 points:
(2,8)(3,14)(4,9)(5,6)(10,25)(11,21)(12,23)(13,22)(15,29)(16,19)(17,18)(20,27)(26,32)(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);(1,2,9,23,6,13,18,21,7,11,17,22,5,12,4,8)(3,10,24,25,14,27,29,31,16,26,30,32,19,28,15,20);(1,3)(2,10)(4,15)(5,19)(6,14)(7,16)(8,20)(9,24)(11,26)(12,28)(13,27)(17,30)(18,29)(21,31)(22,32)(23,25);(2,11)(8,21)(10,26)(12,13)(20,31)(22,23)(25,32)(27,28)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26606);