Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C89 · Schur multiplier (H2) = C23 |
| direct product | C2×C89×#1328 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | #99 |
| commutator | C23 | C22×C8×C89 |
| Frattini, cosocle | C2×M16 | C23×C89 |
| socle | C22×C89 | #324 |
| prime radical | C24×C89 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C22×C4×C89 ◁ C22×M16×C89 ◁ G | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 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);(3,4,6,11,7,12,17,25,8,10,16,22,18,23,29,32)(5,9,14,19,15,21,28,30,13,20,26,24,27,31,34,33);(3,5)(4,9,10,20)(6,14)(7,15)(8,13)(11,24,22,19)(12,21,23,31)(16,26)(17,28)(18,27)(25,33,32,30)(29,34)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4445);