Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C89 · Schur multiplier (H2) = C26 |
| direct product | C89×#10760 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | #117 |
| commutator | C2×C4 | C2×C42×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| socle | C22×C89 | #380 |
| prime radical | C22×#41×C89 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C22×M16×C89 ◁ G | |
1170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 125 points:
(1,2)(3,4)(5,6,9,16,11,15,10,14)(7,17,19,12,21,28,20,27)(8,22,24,30,25,13,23,29)(18,32,34,26,35,31,33,36)(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);(2,4)(5,7,10,20,11,21,9,19)(6,12,16,28,15,27,14,17)(8,18,24,34,25,35,23,33)(13,26,30,32,22,36,29,31);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,29)(15,22)(16,30)(17,31)(19,33)(20,34)(21,35)(27,36)(28,32)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 3748);