Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#22550 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C22×D4 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C22×C79 | C22×C22⋊C4 |
| prime radical | C79×#3480 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C4×Q8×C79 ◁ G | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 95 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 95 points:
(2,6)(7,8)(9,10,13,15,14,16,12,11)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95);(1,2,5,6)(3,7,4,8);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15);(1,4,5,3)(2,7,6,8)5530 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 15538);