Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C79 · Schur multiplier (H2) = C2 |
| direct product | C79×#10609 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #396 |
| commutator, Frattini, cosocle | C2×C42 | C23×C79 |
| prime radical | C23×C79 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 103 points:
(1,2,6,8,5,7,4,3)(9,10,14,16,13,17,12,21)(11,18,22,23,19,24,20,15)(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);(9,11,12,20,13,19,14,22)(10,15,16,18,17,23,21,24);(1,3,5,8)(2,6,7,4)(11,19)(12,14)(16,21)(18,24)2923 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 3597);