Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#19199 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | #532×C79 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 103 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,11)(12,20,21,24)(13,19,15,17)(16,22,18,23)(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);(2,5)(6,7)(9,11,13,19,14,10,15,17)(12,18,22,20,21,16,23,24);(2,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,22)(14,21)(15,23)(17,24)(19,20);(12,21)(16,18)(20,24)(22,23)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 12187);