Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#17353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×C22⋊C4 |
| commutator | C22 | C23×C8×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C22×#41×C79 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C4×Q8×C79 ◁ G | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 103 points:
(1,2,3,5,4,6,7,8)(9,10,12,16,14,17,21,24)(11,18,20,22,13,15,19,23)(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);(9,11,14,13)(10,15,17,18)(12,19,21,20)(16,22,24,23);(9,12,14,21)(10,16,17,24)(11,19,13,20)(15,22,18,23);(9,13,14,11)(10,15,17,18)(12,19,21,20)(16,23,24,22)6478 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 10341);