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×#2082 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C23×C31 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G | |
184 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 184 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) = 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)(32,33,37,42,38,43,36,41)(34,44,47,39,48,55,46,56)(35,49,51,57,52,58,53,40)(45,60,61,63,50,54,62,59);(32,34,38,48)(33,39,43,56)(35,50,52,45)(36,47,37,46)(40,54,57,60)(41,55,42,44)(49,63,58,59)(51,61,53,62);(32,35,36,51,38,52,37,53)(33,40,41,49,43,57,42,58)(34,45,46,61,48,50,47,62)(39,54,44,59,56,60,55,63)713 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C31×#2082, C31×#23652, C31×#24987, … |
| maximal normal subgroup of | C2×C31×#2082, C31×#23652, C31×#24987, … |
| maximal quotient of | C31×#10632, C31×#10633, C31×#10635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 814);