Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#19846 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | #127×C79 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C42×C79 ◁ G | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,5,9)(3,6,11,8)(4,12,14,16)(7,10,15,13)(17,18,22,20)(19,25,27,32)(21,29,31,24)(23,30,28,26)(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);(1,3,5,11)(2,6,9,8)(4,10,14,13)(7,12,15,16)(17,19,22,27)(18,23,20,28)(21,30,31,26)(24,32,29,25);(1,4,5,14)(2,7,9,15)(3,10,11,13)(6,12,8,16);(17,20,22,18)(19,25,27,32)(21,24,31,29)(23,30,28,26)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 12834);