Finite Group DatabasebetaFinite Group Databasebeta| order | 1984 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C2×C4 |
| direct product | C31×#427 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | D4 |
| commutator | C4 | C22×C4×C31 |
| Frattini, cosocle | C2×C4 | C23×C31 |
| socle | C22×C31 | C2×D4 |
| prime radical | C23×C31 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C22×C4×C31 ◁ G | |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)(32,33,34,35)(36,37)(38,45)(39,43)(40,49)(41,44)(42,47)(46,50)(48,51);(36,38,40,47,41,48,39,46)(37,42,43,45,44,50,49,51);(36,39,41,40)(37,43,44,49)(38,46,48,47)(42,45,50,51)868 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1984, 108);