Finite Group DatabasebetaFinite Group Databasebeta| order | 1984 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#508 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C31 | D8 |
| commutator, Frattini, cosocle | C8 | C23×C31 |
| socle | C2×C31 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C31 ◁ C2×C4×C31 ◁ C2×C8×C31 ◁ G | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 3 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)(34,43)(35,40)(36,48)(37,52)(38,42)(39,45)(41,50)(44,59)(46,60)(47,58)(49,54)(51,62)(53,57)(55,63)(56,61);(32,34)(33,39)(35,44)(36,49)(37,53)(38,46)(40,54)(41,57)(42,56)(43,48)(45,50)(47,61)(51,63)(52,55)(58,59)(60,62);(32,35,38,37)(33,40,42,52)(34,44,46,53)(36,47,51,50)(39,54,56,55)(41,48,58,62)(43,59,60,57)(45,49,61,63)682 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1984, 189);