Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#2081 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C22×D4×C31 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G | |
204 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 496 |
A minimal faithful permutation representation — 3 generators on 47 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)(32,33,36,38,37,41,35,40)(34,42,44,47,43,46,45,39);(33,38)(34,43)(35,36)(39,46)(40,41)(42,47);(32,34,35,44,37,43,36,45)(33,39,40,42,41,47,38,46)713 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C31×#2081, C31×#23652, C31×#24987, … |
| maximal normal subgroup of | C2×C31×#2081, C31×#23652, C31×#24987, … |
| maximal quotient of | C31×#10631, C31×#10633, C31×#10634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 813);