Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C22×C4 |
| direct product | C89×#32833 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C89 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | C22×D4×C89 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ #376×C89 ◁ G | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,26)(20,29)(21,25)(22,24)(23,28)(27,32)(30,31)(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);(1,3)(2,5)(4,7)(6,8)(9,11,13,12)(10,14,15,16)(19,27)(20,22)(23,31)(24,29)(26,32)(28,30);(2,6)(5,8)(10,15)(11,12)(17,19)(18,23)(20,27)(21,28)(22,30)(24,32)(25,26)(29,31);(3,7)(5,8)(9,12,13,11)(10,14,15,16)(17,20,21,22)(18,24,25,29)(19,27,28,30)(23,32,26,31)3026 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 25821);