Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33432 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C79 | C23 |
| commutator | C22 | C23×C8×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C22×C79 | C23×C8 |
| prime radical | #525×C79 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C79 ◁ G | |
710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2)(3,4)(5,6,8,13,9,14,18,26,10,12,19,24,20,25,30,34)(7,11,15,22,16,23,27,33,17,21,28,31,29,32,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(2,4);(5,7,8,15,9,16,18,27,10,17,19,28,20,29,30,35)(6,11,13,22,14,23,26,33,12,21,24,31,25,32,34,36);(6,12)(11,21)(13,24)(14,25)(22,31)(23,32)(26,34)(33,36)8848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26420);