Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28 |
| direct product | C2×C89×#3170 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C89 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| socle | C23×C89 | C22×D4 |
| prime radical | C24×C89 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C89 ◁ C22×C42×C89 ◁ G | |
2006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 5 generators on 115 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(3,4)(5,8)(6,7)(9,10)(11,12,15,18)(13,19,22,26)(14,23,20,17)(16,24,25,21);(4,7)(8,10);(11,13,20,24,15,22,14,21)(12,16,23,26,18,25,17,19);(3,5,6,9)(4,8,7,10)7832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53660);