Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C79 · Schur multiplier (H2) = C23×C8 |
| direct product | C2×C79×#1453 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C79 | C23 |
| commutator | C22 | C22×C42×C79 |
| Frattini, cosocle | C42 | C24×C79 |
| socle | C23×C79 | C2×C42 |
| prime radical | C24×C79 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C79 ◁ G | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 105 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79)(80,81);(82,83,85,89,87,88,86,84)(90,91,93,96,94,97,100,103)(92,95,99,102,98,101,104,105);(82,84,86,88,87,89,85,83)(90,92)(91,95)(93,99)(94,98)(96,102)(97,101)(100,104)(103,105);(83,88)(84,89)(92,98)(95,101)(99,104)(102,105)8848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 10309);