Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C2×C79×#1696 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C79 | #121 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C2×C42×C79 ◁ G | |
618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 4 generators on 113 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);(82,83,86,89)(84,90,92,97)(85,94,91,88)(87,95,96,93)(98,99,102,105)(100,106,108,113)(101,109,111,104)(103,110,112,107);(82,84,91,95,86,92,85,93)(83,87,94,97,89,96,88,90);(84,92)(87,96)(90,97)(93,95)(98,100,102,108)(99,103,105,112)(101,110,111,107)(104,113,109,106)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 12137);