Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22×C4 |
| direct product | C31×#1975 |
| 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×C31×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C22×C8×C31 ◁ G | |
292 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 112 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)(33,35)(36,37,40,44,41,45,39,42)(38,46,49,51,47,50,48,43);(32,33,34,35)(37,42,45,44)(38,47)(43,46,51,50)(48,49);(36,38,39,48,41,47,40,49)(37,43,44,46,45,51,42,50)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#7109, C31×#13024, C31×#13037, … |
| maximal normal subgroup of | C31×#7109, C31×#13024, C31×#13037, … |
| maximal quotient of | C31×#9658, C31×#9659, C31×#9660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 707);