Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C2×C4 |
| direct product | C89×#9763 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | #390 |
| commutator | C42 | C22×C4×C89 |
| Frattini, cosocle | C2×C42 | C23×C89 |
| prime radical | C23×C89 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 129 points:
(1,2,3,5,4,6,8,7)(9,10)(11,20,23,31)(12,24)(13,18)(14,29)(15,19)(16,22,32,28)(17,26)(21,30,36,40)(25,33)(27,38)(34,39,37,35)(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)(9,11,13,28,15,23,14,22)(10,16,18,31,19,32,29,20)(12,21,25,39,27,36,26,35)(17,30,24,37,33,40,38,34);(2,6)(5,7)(9,12,14,26,15,27,13,25)(10,17,18,24,19,33,29,38)(11,21,22,35,23,36,28,39)(16,30,31,37,32,40,20,34)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 2751);