Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C25 |
| direct product | C2×C89×#2223 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C24×C89 |
| prime radical | C2×#472×C89 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C22×C4×C89 ◁ C2×#131×C89 ◁ G | |
658 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);(4,10)(5,16)(6,19)(7,8)(11,20)(12,27)(13,24)(14,23)(15,25)(17,32)(18,21)(22,28)(29,34)(30,33);(3,4,11,23,8,14,6,10,9,15,19,24,7,13,20,25)(5,12,26,34,16,28,17,22,18,29,31,27,21,30,32,33);(3,5)(4,12)(6,17)(7,21)(8,16)(9,18)(10,22)(11,26)(13,30)(14,28)(15,29)(19,31)(20,32)(23,34)(24,27)(25,33)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26604);