Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C89 · Schur multiplier (H2) = C25 |
| direct product | C2×C89×#2163 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C89 | C23 |
| commutator | C22 | C23×C8×C89 |
| Frattini, cosocle | C2×C8 | C24×C89 |
| socle | C23×C89 | C22×C8 |
| prime radical | C24×C89 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C89 ◁ G | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 11392 |
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)(5,8)(6,7)(9,10)(11,12,13,17,14,18,19,24,15,16,20,22,21,23,25,26);(4,7)(8,10);(3,5,6,9)(4,8,7,10)(12,16)(17,22)(18,23)(24,26)9968 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26404);