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) = C22 |
| direct product | C31×#443 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C31 | D4 |
| commutator | C4 | C22×C4×C31 |
| Frattini, cosocle | C2×C4 | C23×C31 |
| socle | C2×C31 | C4×D4 |
| prime radical | C31×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C31 ◁ C2×C8×C31 ◁ G | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 496 |
A minimal faithful permutation representation — 3 generators on 47 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)(32,33,36,40,37,34,41,39)(35,38,43,46,44,42,47,45);(32,34)(33,37)(35,42)(36,39)(38,44)(40,41)(43,45)(46,47);(32,35,36,43,37,44,41,47)(33,38,40,46,34,42,39,45)868 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1984, 124);