Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#19110 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #121 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| socle | C22×C79 | C2×#121 |
| prime radical | C79×#3100 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ G | |
716 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)(3,5)(4,6)(7,8)(9,10)(11,18)(12,15)(13,17)(14,19)(16,21)(20,22)(23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(2,5)(6,7)(9,11,12,19,13,20,21,24)(10,14,16,18,17,23,15,22);(2,6)(4,8)(5,7);(9,12,13,21)(10,15,17,16)(11,19,20,24)(14,22,23,18)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 12098);