Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#60496 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| prime radical | C2×H(5,𝔽2)×C89 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C4×#41×C89 ◁ G | |
2054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 5 generators on 113 points:
(1,2,4,6)(3,7,8,5)(9,10,12,14)(11,16,18,21)(13,15,20,17)(19,22,24,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(1,3,4,8)(2,5,6,7)(10,14)(15,17)(16,21)(22,23);(10,15)(13,20)(14,17)(16,22)(19,24)(21,23);(9,11)(10,16)(12,18)(13,19)(14,21)(15,22)(17,23)(20,24);(11,18)(16,21)(19,24)(22,23)5429 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53484);