Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17547 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#17549 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | #121×C79 | C23 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C4×Q8×C79 ◁ G | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 103 points:
(2,5,6,7)(9,10,14,13)(11,18,22,24)(12,16,23,17)(15,19,20,21)(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);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,20)(14,22)(16,18)(17,24)(21,23);(9,12,14,23)(10,16,13,17)(11,19,22,21)(15,18,20,24);(9,13,14,10)(11,20,22,15)(12,16,23,17)(18,21,24,19)4819 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 10537);