Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#13013 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | C2×D8 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C22×D8×C79 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 103 points:
(1,2,4,3)(5,6,7,8)(9,10,14,20,24,23,18,11,16,22,13,19,15,21,17,12)(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);(1,3,4,2)(6,8)(9,11,16,12)(10,17,22,18)(13,21,14,23)(15,19,24,20);(9,12,16,11)(10,18,22,17)(13,23,14,21)(15,20,24,19)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6001);