Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C79 · Schur multiplier (H2) = C28 |
| direct product | C4×#560×C79 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C24 |
| commutator | C22 | C24×C4×C79 |
| Frattini, cosocle | C23 | C25×C79 |
| socle | C23×C79 | C25 |
| prime radical | C2×#560×C79 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ G | |
2568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 5 generators on 99 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79)(80,81,82,83);(84,85)(86,90)(87,89)(88,91)(92,93)(94,96)(95,97)(98,99);(84,86)(85,88)(87,91)(89,90)(92,94)(93,96)(95,98)(97,99);(86,91)(88,90)(93,97)(94,98);(85,89)(86,91)(93,97)(96,99)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 27339);