Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C79×#1454 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C79 | C23 |
| commutator | C22 | C22×C42×C79 |
| Frattini, cosocle | C42 | C24×C79 |
| socle | C23×C79 | C2×C42 |
| prime radical | C24×C79 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C79 ◁ G | |
802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 10112 |
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);(82,83,85,89,86,88,92,95)(84,87,91,93,90,94,96,97)(98,99)(100,103)(101,105)(102,104)(106,110)(107,109)(108,111)(112,113);(82,84)(83,87)(85,91)(86,90)(88,94)(89,93)(92,96)(95,97)(98,100,101,106,102,107,108,112)(99,103,105,110,104,109,111,113);(83,88)(84,90)(89,95)(91,96)(99,104)(103,109)(105,111)(110,113)8848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 10310);