Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#18517 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| socle | C22×C79 | C2×#120 |
| prime radical | D4×#41×C79 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ G | |
1204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 103 points:
(1,2,4,7)(3,5,8,6)(9,10,14,18)(11,17,21,24)(12,15,13,16)(19,22,20,23)(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)(3,8)(6,7)(10,15)(12,13)(16,18)(17,22)(19,20)(23,24);(10,16)(12,13)(15,18)(17,23)(19,20)(22,24);(9,11,14,21)(10,17,18,24)(12,19,13,20)(15,22,16,23)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11505);