Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#1884 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C31 | #120 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C2×C31 | #386 |
| prime radical | C31×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C31 ◁ C22×C4×C31 ◁ G | |
332 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 108 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)(34,41,42,45)(35,38,43,44)(36,39)(37,40)(46,47);(32,34,36,44,37,42,46,38)(33,35,39,45,40,43,47,41);(32,35)(33,38)(34,39)(36,45)(37,43)(40,44)(41,46)(42,47)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#7482, C31×#7484, C31×#7486, … |
| maximal normal subgroup of | C31×#7482, C31×#7484, C31×#7486, … |
| maximal quotient of | C31×#8957, C31×#8958, C31×#8959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 616);