Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#19183 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C89 |
| prime radical | C89×#3106 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,31)(30,32)(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);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16);(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26);(3,10)(6,15)(9,16)(11,14)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)3827 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 12171);