Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#18723 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C89 |
| prime radical | C89×#3548 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 121 points:
(2,7)(4,5)(8,14)(9,13)(10,12)(15,16)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,28,32,29)(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);(1,2,6,9)(3,8,11,16)(4,7,5,13)(10,15,12,14)(17,19,22,30)(18,23,26,32)(20,28,21,29)(24,31,25,27);(1,3)(2,8)(4,12)(5,10)(6,11)(7,14)(9,16)(13,15)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29);(1,4,6,5)(2,7,9,13)(3,10,11,12)(8,15,16,14)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27)3560 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 11711);