Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C24 |
| direct product | C2×C79×#1381 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | D8 |
| commutator | C8 | C2×C42×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C23×C79 | #107 |
| prime radical | C24×C79 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ C23×C4×C79 ◁ C23×C8×C79 ◁ G | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 117 points:
(1,2)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(7,8,11,16,13,19,25,18)(9,20,22,34,24,35,36,29)(10,17,12,15,26,32,27,33)(14,23,28,37,31,38,30,21);(3,4,5,6)(7,9,12,23,11,22,26,37,13,24,27,38,25,36,10,21)(8,14,17,29,18,30,33,35,19,31,32,34,16,28,15,20)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5546);