Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C23×C8 |
| direct product | C2×C79×#1708 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | #127 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | C2×#127×C79 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ 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);(83,86)(84,88)(87,89)(90,91)(92,98)(93,101)(94,96)(95,99)(97,102)(100,104)(103,105);(90,92,93,99,94,100,102,105)(91,95,97,98,96,103,101,104);(82,83,84,86,85,87,88,89)(91,96)(95,103)(97,101)(98,104)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 12583);