Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C79 · Schur multiplier (H2) = C28 |
| direct product | C4×#564×C79 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C24 |
| commutator | C22 | C24×C4×C79 |
| Frattini, cosocle | C23 | C25×C79 |
| socle, prime radical | C23×C79 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ G | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 5 generators on 115 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)(86,89)(87,94)(88,91)(90,96)(92,99)(93,98)(95,97)(100,101,104,107)(102,108,109,105)(103,111,113,106)(110,114,112,115);(84,86,88,93)(85,89,91,98)(87,95,96,92)(90,99,94,97)(100,102)(101,105)(103,112)(104,109)(106,115)(107,108)(110,113)(111,114);(84,87,88,96)(85,90,91,94)(86,92,93,95)(89,97,98,99)(102,109)(105,108)(110,112)(114,115);(85,91)(89,98)(90,94)(97,99)(100,103,104,113)(101,106,107,111)(102,110,109,112)(105,114,108,115)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 27343);