Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C19 · Schur multiplier (H2) = C1 |
| direct product | C19×#368 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | D8 |
| commutator | C8 | C2×C4×C19 |
| Frattini, cosocle | C2×C8 | C22×C19 |
| socle | C2×C19 | #107 |
| prime radical | C22×C19 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C2×C4×C19 ◁ C2×C8×C19 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 2 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,29,26,30,25,40)(22,31,34,46,36,47,35,50)(23,37,38,44,39,51,32,28)(27,41,42,49,43,48,45,33);(20,22,32,48,25,35,39,49,26,36,38,41,24,34,23,33)(21,27,37,46,29,42,44,47,30,43,51,50,40,45,28,31)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 49);