Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16×C79 · Schur multiplier (H2) = C1 |
| direct product | C79×#7459 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C79 | D8 |
| commutator | C8 | C2×C16×C79 |
| Frattini, cosocle | C4×C16 | C22×C79 |
| socle | C2×C79 | #1370 |
| prime radical | C22×C79 | #364 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C79 ◁ C2×C16×C79 ◁ C4×C16×C79 ◁ G | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 2 generators on 111 points:
(1,2,3,10,6,9,16,23,5,12,15,28,20,25,31,24,7,13,17,29,21,26,32,22,19,11,30,27,4,18,14,8)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(1,3,6,16,5,15,20,31,7,17,21,32,19,30,4,14)(2,8,18,27,11,22,26,29,13,24,25,28,12,23,9,10)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 447);