Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C89 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C4×#124×C89 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C89 | C23 |
| commutator | C22 | C24×C4×C89 |
| Frattini, cosocle | C23 | C25×C89 |
| socle | C24×C89 | C24 |
| prime radical | C23×D4×C89 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C89 ◁ G | |
3790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 5 generators on 107 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(3,4,5,6);(8,9)(11,12)(13,16)(14,15)(17,18);(7,8,10,9)(12,15)(16,18);(7,9,10,8)(11,13,14,17)(12,16,15,18)9968 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 27020);