Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12977 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C79 | C23 |
| commutator | C23 | C22×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C23×C79 | C22×C8 |
| prime radical | C22×D4×C79 | C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C79 ◁ G | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 103 points:
(1,2,3,4)(5,6)(7,8)(9,10,12,15,13,16,19,22,11,14,17,20,18,21,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(2,4)(9,10,12,15,13,16,19,22,11,14,17,20,18,21,23,24);(6,8)(10,14)(15,20)(16,21)(22,24)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5965);