Finite Group DatabasebetaFinite Group Databasebeta| order | 1984 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C31 · Schur multiplier (H2) = C1 |
| direct product | C31×#368 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C31 | D8 |
| commutator | C8 | C2×C4×C31 |
| Frattini, cosocle | C2×C8 | C22×C31 |
| socle | C2×C31 | #107 |
| prime radical | C22×C31 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C31 ◁ C2×C4×C31 ◁ C2×C8×C31 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 2 generators on 63 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)(32,33,36,41,38,42,37,52)(34,43,46,58,48,59,47,62)(35,49,50,56,51,63,44,40)(39,53,54,61,55,60,57,45);(32,34,44,60,37,47,51,61,38,48,50,53,36,46,35,45)(33,39,49,58,41,54,56,59,42,55,63,62,52,57,40,43)682 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1984, 49);