Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#1950 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C2×D4 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C22×C31 | C2×C4⋊C4 |
| prime radical | C23×C31 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G | |
200 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 1984 |
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)(36,37,40,43,41,44,39,48)(38,45,46,42,47,50,49,51);(36,38,40,46,41,47,39,49)(37,42,43,50,44,51,48,45);(32,33,34,35)1178 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#8906, C31×#8907, C31×#8908, … |
| maximal normal subgroup of | C31×#8906, C31×#8907, C31×#8908, … |
| maximal quotient of | C31×#9601, C31×#9602, C31×#9603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 682);