Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C22×C42 |
| direct product | C79×#12581 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #99 |
| commutator | C2×C4 | C2×C42×C79 |
| Frattini, cosocle | C22×D4 | C23×C79 |
| socle | C22×C79 | C2×#99 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C22×C4×C79 ◁ C22×#41×C79 ◁ G | |
854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 99 points:
(5,6)(7,11)(8,12,16,10)(9,13)(14,18,20,17)(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);(1,2,3,4)(6,10,13,12)(11,17,19,18);(5,7,9,15)(6,11,13,19)(8,14,16,20)(10,17,12,18)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5569);