Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C79 · Schur multiplier (H2) = C24×C4 |
| direct product | C79×#20485 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C79 | C23 |
| commutator | C22 | C22×C42×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C23×C79 | C23×C4 |
| prime radical | C22×D4×C79 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C79 ◁ G | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 103 points:
(5,6)(7,8)(9,10,12,15,13,16,20,23)(11,17,19,24,18,14,21,22)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(1,2,4,3)(6,8)(11,18)(14,17)(19,21)(22,24);(1,3,4,2)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,21)(23,24);(9,10,12,15,13,16,20,23)(11,14,19,22,18,17,21,24)8848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 13473);