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×#1977 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | #121 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C22×D4×C31 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C22×C4×C31 ◁ G | |
296 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 296 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)(36,37,41,45,42,44,40,38)(39,47,49,51,50,46,48,43);(32,33,34,35)(36,38,41,44,42,45,40,37)(39,46,49,51,50,47,48,43);(36,39,40,48,42,50,41,49)(37,43,45,47,44,51,38,46)992 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C31×#1977, C31×#21248, C31×#21249, … |
| maximal normal subgroup of | C2×C31×#1977, C31×#21248, C31×#21249, … |
| maximal quotient of | C31×#9667, C31×#9672, C31×#9723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 709);