Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24×C4 |
| direct product | C79×#25624 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C79×#3151 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 103 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,12)(11,15,20,18)(13,21,16,17)(19,24,22,23)(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)(9,11,14,20)(10,15,12,18)(13,22,16,19)(17,24,21,23);(9,12,13,21,14,10,16,17)(11,18,19,24,20,15,22,23);(1,3,4,8)(2,6,5,7)3397 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 18612);