Finite Group DatabasebetaFinite Group Databasebeta| order | 1984 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C31 · Schur multiplier (H2) = C2×C4 |
| direct product | C31×#405 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C31 | C22 |
| commutator | C2 | C2×C42×C31 |
| Frattini, cosocle | C2×C4 | C23×C31 |
| socle | C22×C31 | C22×C4 |
| prime radical | C23×C31 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C31 ◁ G | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 3 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)(36,37,39,42,40,43,46,49)(38,41,45,48,44,47,50,51);(32,33,34,35)(36,38)(37,41)(39,45)(40,44)(42,48)(43,47)(46,50)(49,51);(38,44)(41,47)(45,50)(48,51)1240 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1984, 86);