Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C25 |
| direct product | C2×C89×#2080 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| socle | C22×C89 | #397 |
| prime radical | C22×#41×C89 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C23×C4×C89 ◁ G | |
906 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)(5,13,16,26)(6,10)(7,11)(8,12)(9,15,23,20)(14,29,17,28)(18,24)(19,25)(21,27)(22,31,33,30)(32,34);(3,5)(4,9)(6,17)(7,20)(8,16)(10,22)(11,26)(12,23)(13,27)(14,19)(15,21)(18,30)(24,29)(25,33)(28,34)(31,32);(3,6,7,18,8,19,21,32)(4,10,11,24,12,25,27,34)(5,14,15,30,16,17,20,31)(9,22,13,28,23,33,26,29)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 16636);