Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C89 · Schur multiplier (H2) = C23 |
| direct product | D4×C32×C89 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C89 | C22 |
| commutator | C2 | C22×C32×C89 |
| Frattini, cosocle | C2×C16 | C23×C89 |
| socle | C22×C89 | C22×C16 |
| prime radical | C2×D4×C89 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C89 ◁ G | |
214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 3 generators on 125 points:
(5,6,7,11,8,12,15,21,9,13,16,22,18,24,27,31,10,14,17,23,19,25,28,32,20,26,29,33,30,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,2)(3,4);(2,4)14240 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 6632);