Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C79×#1281 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C22⋊C4 |
| commutator | C22 | C2×C4×C8×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| socle | C23×C79 | #95 |
| prime radical | C24×C79 | C42 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ C22×C42×C79 ◁ G | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79)(80,81);(82,83,84,86,85,87,88,89)(90,91)(92,98,99,95)(93,97)(94,96)(100,104,105,102)(101,103);(90,92,93,100,94,99,101,105)(91,95,97,102,96,98,103,104)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 905);