Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C24 |
| direct product | C2×C79×#1394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | #99 |
| commutator | C2×C4 | C2×C42×C79 |
| Frattini, cosocle | C22×Q8 | C23×C79 |
| prime radical | C24×C79 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C24×C79 ◁ C23×Q8×C79 ◁ G | |
818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 101 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(7,8,10,14,11,15,19,22)(9,13,16,21,17,12,18,20);(3,4,5,6)(8,12,15,13)(14,21,22,20)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5572);