Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C89×#1937 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C89 | C2×D4 |
| commutator | C2×C4 | C23×C4×C89 |
| Frattini, cosocle | C22×C4 | C24×C89 |
| socle | C23×C89 | C4×D4 |
| prime radical | C22×#41×C89 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C89 ◁ C23×C8×C89 ◁ G | |
1114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2)(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);(3,4)(5,6)(7,8,11,15,12,16,10,13)(9,17,19,14,20,21,18,22);(8,13,16,15)(14,17,22,21);(4,6)(7,9,10,18,12,20,11,19)(8,14,15,21,16,22,13,17)7832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 14084);