Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C25 |
| direct product | C2×C89×#2061 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C89 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | C24×C89 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C23×C4×C89 ◁ G | |
1130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 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);(3,4,7,9)(5,10,6,8)(11,12,14,18)(13,20,21,16)(15,17,23,19)(22,26,24,25);(3,5,7,6)(4,8,9,10)(11,13,14,21)(12,16,18,20)(15,24,23,22)(17,26,19,25);(3,6,7,5)(4,8,9,10)(12,17)(15,23)(16,25)(18,19)(20,26)(22,24)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 16547);