Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C24 |
| direct product | C2×C89×#2166 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | #117 |
| commutator | C2×C4 | C23×C4×C89 |
| Frattini, cosocle | C2×C8 | C24×C89 |
| socle | C22×C89 | #432 |
| prime radical | C24×C89 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C2×#97×C89 ◁ G | |
538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 123 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(3,4,6,12,7,13,20,29,8,11,19,27,21,10,24,26)(5,14,16,28,17,25,31,34,15,9,18,23,30,22,32,33);(3,5)(4,9)(6,18)(7,17)(8,15)(10,25)(11,14)(12,28)(13,22)(16,19)(20,32)(21,30)(23,27)(24,31)(26,33)(29,34);(4,10,11,13)(5,15)(6,19)(9,22,14,25)(12,29,27,26)(17,30)(20,24)(23,34,28,33)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26413);