Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#12616 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×C42 |
| commutator | C2×C4 | C2×C42×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C2×#97 |
| prime radical | C24×C79 | M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 99 points:
(1,2)(3,4)(5,6,8,13,9,11,17,18,10,14,16,20,7,12,15,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(2,4)(6,11,14,12)(8,16)(13,19,20,18)(15,17);(5,6,8,13,9,11,17,18,10,14,16,20,7,12,15,19)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5604);