Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#23539 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C23×C79 | C22×D4 |
| prime radical | C22×#135×C79 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ C22×C42×C79 ◁ G | |
1494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 103 points:
(1,2)(3,4)(10,15)(12,13)(16,22)(17,18)(19,20)(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);(5,6)(7,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,23)(20,22)(21,24);(2,4)(6,8)(9,10,12,15,14,18,13,17)(11,16,19,22,21,24,20,23);(9,11,14,21)(10,16,18,24)(12,19,13,20)(15,22,17,23)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16527);