Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#21571 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C24 |
| commutator | C23 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C23×C79 | C23×C4 |
| prime radical | C22×D4×C79 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ G | |
872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 107 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,20,18,21,26,16)(15,19,24,23,22,27,28,25)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(6,9)(10,12)(13,15)(14,19)(16,25)(17,24)(18,22)(20,23)(21,27)(26,28);(2,4)(5,7,8,11)(6,10,9,12)(15,22)(19,27)(23,25)(24,28);(13,16,17,14,18,20,26,21)(15,23,24,27,22,25,28,19)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 14559);