Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C2×C4 |
| direct product | C31×#1938 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C2×D4 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C22×C31 | C4×D4 |
| prime radical | C2×#41×C31 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C22×C8×C31 ◁ G | |
222 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(32,33)(34,35)(36,37,40,44,41,45,39,42)(38,46,49,43,47,50,48,51);(37,42,45,44)(38,47)(43,50,51,46)(48,49);(33,35)(36,38,39,48,41,47,40,49)(37,43,44,50,45,51,42,46)992 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C31×#1938, C31×#21098, C31×#21099, … |
| maximal normal subgroup of | C2×C31×#1938, C31×#21098, C31×#21099, … |
| maximal quotient of | C31×#9555, C31×#9556, C31×#9557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 670);