Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C32×C79 · Schur multiplier (H2) = C4 |
| direct product | C79×#7331 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C16×C79 | C22 |
| commutator | C2 | C4×C32×C79 |
| Frattini, cosocle | C4×C16 | C22×C79 |
| socle, prime radical | C22×C79 | C4×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C16×C79 ◁ G | |
118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 2 generators on 119 points:
(1,2,3,6,4,5,7,8)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(2,5)(6,8)(9,10,11,15,12,16,19,25,13,17,20,26,22,28,31,35,14,18,21,27,23,29,32,36,24,30,33,37,34,38,39,40)12640 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 319);