Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C79×#22835 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C24 |
| commutator | C22 | C23×C8×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C23×C79 | C23×C4 |
| prime radical | C22×D4×C79 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ G | |
924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 103 points:
(1,2,3,5,4,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(10,13)(11,15)(17,19,21,20)(18,22,23,24);(10,13)(14,16)(17,20,21,19)(18,22,23,24);(9,11,12,15)(10,14,13,16)7900 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 15823);