Finite Group DatabasebetaFinite Group Databasebeta| order | 1984 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C31 · Schur multiplier (H2) = C25 |
| direct product | C31×#519 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | C24 |
| commutator | C2 | C23×C4×C31 |
| Frattini, cosocle | C22 | C24×C31 |
| prime radical | C23×C31 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ G | |
266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 4 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)(37,41)(39,47)(42,44)(45,51);(36,37,40,41)(38,44,46,42)(39,43,47,48)(45,50,51,49);(36,38,40,46)(37,42,41,44)(39,45,47,51)(43,49,48,50);(36,39)(37,43)(38,45)(40,47)(41,48)(42,49)(44,50)(46,51)1054 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1984, 200);