Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C24×C4 |
| direct product | C89×#25594 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | C2×AGL(1,ℤ/8)×C89 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 129 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,18)(11,15,20,29)(12,16,23,32)(13,25,27,17)(19,35,36,28)(21,34,26,30)(22,31,37,40)(24,39,38,33)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(2,5)(6,7)(9,11)(10,15)(12,22)(13,26)(14,20)(16,31)(17,34)(18,29)(19,24)(21,27)(23,37)(25,30)(28,33)(32,40)(35,39)(36,38);(9,12)(10,16)(11,19,20,36)(13,24)(14,23)(15,28,29,35)(17,33)(18,32)(21,22,26,37)(25,39)(27,38)(30,31,34,40);(1,3,4,8)(2,6,5,7)(12,23)(16,32)(19,36)(22,37)(24,38)(28,35)(31,40)(33,39)3560 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 18582);