Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C79 · Schur multiplier (H2) = C213 |
| direct product | C2×C79×#3531 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C24 |
| commutator, Frattini, cosocle | C22 | C26×C79 |
| socle | C23×C79 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ G | |
6554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 6 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)(84,87)(85,92)(86,89)(88,94)(90,97)(91,96)(93,95)(98,99)(100,106)(101,105)(102,104)(103,107)(108,112)(109,111)(110,113);(82,84,86,91)(83,87,89,96)(85,93,94,90)(88,97,92,95)(98,100)(99,103)(101,108)(102,107)(104,106)(105,110)(109,113)(111,112);(82,85,86,94)(83,88,89,92)(84,90,91,93)(87,95,96,97)(100,107)(103,106)(108,113)(110,112);(83,89)(87,96)(88,92)(95,97)(99,104)(100,107)(105,111)(108,113);(98,101,102,109)(99,105,104,111)(100,108,107,113)(103,110,106,112)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 55681);