Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C210 |
| direct product | C2×C79×#2397 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C79 | C23 |
| commutator, Frattini, cosocle | C23 | C25×C79 |
| socle | C24×C79 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C79 ◁ G | |
5150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 5 generators on 97 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)(90,91,93,94)(92,95,96,97);(83,85)(86,87)(88,89)(91,94)(95,97);(83,85)(87,89);(90,92,93,96)(91,95,94,97)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 27532);