Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C25 |
| direct product | C2×C89×#2176 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C89 | C2×D8 |
| commutator | C8 | C23×C4×C89 |
| Frattini, cosocle | C2×C8 | C24×C89 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C2×C42×C89 ◁ C2×C4×C8×C89 ◁ G | |
670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 22784 |
A minimal faithful permutation representation — 4 generators on 127 points:
(1,2)(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);(3,4,6,5)(7,8)(9,17)(10,21)(11,26)(12,14)(13,18)(15,24)(16,23)(19,37)(20,34)(22,32)(25,33)(27,31)(28,36)(29,38)(30,35);(7,9,12,20)(8,13,14,28)(10,22,25,38)(11,27,23,19)(15,31,30,37)(16,32,26,29)(17,33,34,21)(18,35,36,24);(3,5,6,4)(8,14)(13,28)(15,30)(16,26)(17,34)(21,33)(29,32)(31,37)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26437);