Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#11306 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | #124 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| socle | C22×C89 | #400 |
| prime radical | C24×C89 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C23×C4×C89 ◁ G | |
530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 125 points:
(5,6,11,14,10,15,9,16)(7,17,21,27,20,28,19,12)(8,13,24,30,25,31,23,32)(18,29,22,35,34,36,33,26)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,2)(3,4)(5,7,11,21,10,20,9,19)(6,12,16,28,15,27,14,17)(8,22,23,18,25,33,24,34)(13,29,30,35,31,36,32,26);(2,4)(5,8,9,23,10,25,11,24)(6,13,14,30,15,31,16,32)(7,18,19,33,20,34,21,22)(12,26,17,29,27,35,28,36)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4294);