Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C79×#7572 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| prime radical | C23×D4×C79 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C22×C42×C79 ◁ G | |
884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 103 points:
(1,2,3,5,4,6,7,8)(9,10)(11,15,16,13)(12,14)(17,18)(19,21)(20,22)(23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(9,11)(10,13)(12,16)(14,15)(18,21)(19,24)(22,23);(10,14)(13,15)(18,22)(21,23)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 560);