Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#23590 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C22×C79 | C2×#120 |
| prime radical | #537×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ G | |
1074 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,15)(13,17)(14,23)(16,20)(18,22)(21,24)(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)(6,8)(9,11,13,12)(10,15,17,19)(14,21,22,20)(16,18,24,23);(9,11,13,12)(10,16,17,24)(14,21,22,20)(15,23,19,18);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,17,19)(14,21,22,20)(16,23,24,18)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16578);