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×#518 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | C24 |
| commutator | C2 | C23×C4×C31 |
| Frattini, cosocle | C22 | C24×C31 |
| prime radical | C22×D4×C31 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ G | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 496 |
A minimal faithful permutation representation — 4 generators on 43 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)(36,37)(38,42)(39,40)(41,43);(37,40)(41,42);(36,37)(38,41)(39,40)(42,43);(36,38)(37,41)(39,43)(40,42)1054 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1984, 199);