Finite Group DatabasebetaFinite Group Databasebeta| order | 1984 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#504 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C31 | C22 |
| commutator | C2 | C22×C8×C31 |
| Frattini, cosocle | C8 | C23×C31 |
| socle | C2×C31 | C22×C8 |
| prime radical | #41×C31 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×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)(32,33,35,39,36,40,45,51,37,41,46,52,47,53,57,61)(34,38,43,49,44,50,56,60,42,48,54,58,55,59,62,63);(32,34)(33,38)(35,43)(36,44)(37,42)(39,49)(40,50)(41,48)(45,56)(46,54)(47,55)(51,60)(52,58)(53,59)(57,62)(61,63);(34,42)(38,48)(43,54)(44,55)(49,58)(50,59)(56,62)(60,63)1240 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1984, 185);