Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C2×C79×#1683 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | #121 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | C22×D4×C79 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C2×C42×C79 ◁ G | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 105 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)(84,86)(85,87)(88,89)(90,91,94,97)(92,98,101,105)(93,102,99,96)(95,103,104,100);(83,86)(87,88)(90,92,99,103,94,101,93,100)(91,95,102,105,97,104,96,98);(83,87)(85,89)(86,88)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 12097);