Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#17890 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C79 | C2×D4 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C42 | C24×C79 |
| socle | C22×C79 | C2×C4×D4 |
| prime radical | C2×#135×C79 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 103 points:
(1,2,5,7,4,6,8,3)(9,10)(11,18)(12,16)(13,15)(14,20)(17,22)(19,23)(21,24)(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);(2,6)(3,7)(9,11)(10,14)(12,19)(13,21)(15,23)(16,18)(17,24)(20,22);(1,3,5,2,4,7,8,6)(10,15)(14,23)(16,17)(18,24);(9,12,13,22)(10,16,15,17)(11,19,21,20)(14,18,23,24)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 10878);