Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#19911 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C89 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C42×C89 ◁ G | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2,5,8)(3,9,10,16)(4,13,14,7)(6,11,15,12)(17,18,21,25)(19,22,27,31)(20,23,28,24)(26,29,32,30)(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)(2,6)(4,11)(5,10)(7,16)(8,15)(9,13)(12,14)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(3,10)(6,15)(9,16)(11,12)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(3,11)(4,14)(6,16)(7,13)(9,15)(10,12)3827 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 12899);