Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C89 · Schur multiplier (H2) = C23 |
| direct product | C89×#10224 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C89 |
| prime radical | C23×C89 | #127 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2,7,11)(3,12,4,15)(5,10,6,9)(8,14,16,13)(19,26)(20,21)(23,27)(24,30)(28,29)(31,32)(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);(1,3,7,4)(2,8,11,16)(5,14,6,13)(9,15,10,12)(17,18)(19,27)(20,24)(21,30)(22,25)(23,29)(26,31)(28,32);(1,4,7,3)(2,8,11,16)(5,13,6,14)(9,15,10,12)(17,19)(18,23)(20,29)(21,26)(22,28)(24,32)(25,31)(27,30)3026 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 3212);