Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#21481 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator | C23 | C23×C4×C79 |
| Frattini, cosocle | C24 | C24×C79 |
| prime radical | C2×H(5,𝔽2)×C79 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C23×C4×C79 ◁ G | |
1494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 99 points:
(1,2,4,3)(5,6)(7,10)(8,11)(9,12)(13,19)(14,17)(15,18)(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)(6,10)(8,13)(9,14)(11,16)(12,17)(15,20)(18,19);(5,8)(6,11)(7,13)(9,15)(10,16)(12,18)(14,20)(17,19);(1,3,4,2)(6,12)(10,17)(11,18)(16,19)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 14469);