Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C79×#2164 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C79 | C23 |
| commutator | C22 | C23×C8×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C23×C79 | C22×C8 |
| prime radical | C24×C79 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C79 ◁ G | |
610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 121 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)(84,87)(85,86)(88,89)(90,91,93,97,94,98,103,109,95,99,104,110,105,111,115,119)(92,96,101,107,102,108,114,118,100,106,112,116,113,117,120,121);(83,86)(87,89)(90,92)(91,96)(93,101)(94,102)(95,100)(97,107)(98,108)(99,106)(103,114)(104,112)(105,113)(109,118)(110,116)(111,117)(115,120)(119,121);(82,84,85,88)(83,87,86,89)(92,100)(96,106)(101,112)(102,113)(107,116)(108,117)(114,120)(118,121)8848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26405);