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×#23451 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C23×C79 | C22×D4 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ C22×C42×C79 ◁ G | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 107 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,22)(16,19)(17,21)(18,24)(20,26)(23,27)(25,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(1,2)(3,4)(5,7,8,12)(6,9,10,11);(2,4)(13,15)(14,18)(16,23)(17,25)(19,22)(20,28)(21,27)(24,26);(13,16,17,26)(14,19,21,20)(15,23,25,24)(18,22,27,28)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16439);