Finite Group DatabasebetaFinite Group Databasebeta| order | 1984 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C31 · Schur multiplier (H2) = C2 |
| direct product | C31×#350 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C31 | D4 |
| commutator | C4 | C2×C8×C31 |
| Frattini, cosocle | C2×C8 | C22×C31 |
| socle | C2×C31 | C22⋊C8 |
| prime radical | #41×C31 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C31 ◁ C2×C8×C31 ◁ G | |
56 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)(32,33,36,41,37,42,51,59,38,43,52,60,35,40,49,57)(34,44,46,61,45,55,62,58,47,56,50,63,48,39,53,54);(32,34)(33,39)(35,48)(36,50)(37,45)(38,47)(40,56)(41,58)(42,44)(43,55)(46,52)(49,62)(51,53)(54,60)(57,61)(59,63)868 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1984, 31);