Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#60307 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C89 | C24 |
| commutator | C22 | C24×C4×C89 |
| Frattini, cosocle | C2×C4 | C25×C89 |
| socle | C22×C89 | C24×C4 |
| prime radical | C89×#3529 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C89 ◁ G | |
1820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 5 generators on 129 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,16,24,29)(11,15,21,27,19,25,32,36)(12,17,22,30,23,28,34,38)(20,26,33,37,31,35,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,25)(17,26)(18,27)(22,33)(23,31)(24,32)(28,35)(29,36)(30,37)(34,39)(38,40);(3,8)(5,7)(10,16)(11,19)(17,28)(18,29)(20,31)(21,32)(30,38)(33,39);(2,6)(5,7)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40);(9,12,13,22,14,23,24,34)(10,17,18,30,16,28,29,38)(11,20,21,33,19,31,32,39)(15,26,27,37,25,35,36,40)6764 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53295);