Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C24 |
| direct product | C2×C79×#1326 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | #99 |
| commutator | C23 | C22×C8×C79 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C79 |
| prime radical | C25×C79 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C24×C79 ◁ C22×C22⋊C4×C79 ◁ G | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 97 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)(92,96,97,94)(93,95);(91,94,95,96)(92,97)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 4439);