Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#23609 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C23×C79 | C22×D4 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ C22×C42×C79 ◁ G | |
1138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 103 points:
(1,2,3,4)(10,16)(11,18)(13,14)(17,23)(19,24)(20,21)(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);(5,6)(7,8)(9,10,13,16,15,18,14,11)(12,17,20,23,22,24,21,19);(2,4)(6,8)(9,11)(10,14)(12,19)(13,18)(15,16)(17,21)(20,24)(22,23);(9,12,15,22)(10,17,18,24)(11,19,16,23)(13,20,14,21)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16597);