Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C27 |
| direct product | C89×#23424 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C89 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | C22×AGL(1,ℤ/8)×C89 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C2×#117×C89 ◁ G | |
2098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 113 points:
(1,2)(3,4)(10,14)(11,17)(12,20)(16,21)(19,22)(23,24)(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);(2,4)(5,6)(7,8)(9,10)(11,18)(12,21)(13,16)(14,20)(15,19)(17,24)(22,23);(6,8)(9,11,13,19)(10,15,16,18)(12,22,20,17)(14,23,21,24);(10,16)(14,21)(15,18)(23,24)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 16412);