Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#24467 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C79×#3488 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C23×C4×C79 ◁ G | |
1486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 99 points:
(5,6,11,15,12,16,10,20)(7,17,9,13,18,14,8,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(1,2)(3,4)(5,7)(6,13)(8,10)(9,11)(12,18)(14,15)(16,19)(17,20);(2,4)(5,8,12,9)(6,14,16,17)(7,11,18,10)(13,20,19,15);(5,9,12,8)(6,14,16,17)(7,11,18,10)(13,15,19,20)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 17455);