Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C79 · Schur multiplier (H2) = C26 |
| direct product | C22×#335×C79 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C79 | D4 |
| commutator | C4 | C23×C8×C79 |
| Frattini, cosocle | C42 | C24×C79 |
| socle, prime radical | C24×C79 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C79 ◁ C22×C42×C79 ◁ G | |
942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 4 generators on 99 points:
(1,2)(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);(3,4);(5,6,7,9,8,10,11,12)(14,17)(15,18)(19,20);(13,14,18,20,16,19,15,17)8848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 10772);