Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#13445 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C2×#509×C79 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 99 points:
(1,2)(3,4)(5,6)(7,18)(8,14)(9,13)(10,12)(11,17)(15,19)(16,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(5,7)(6,12)(8,11)(9,10)(14,18)(16,17)(19,20);(2,4)(5,6,8,13,20,12,19,18,11,17,7,15,10,16,9,14)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6433);