Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22×C4 |
| direct product | C31×#2126 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C31 | #99 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C2×D4 | C23×C31 |
| socle | C2×C31 | C2×#99 |
| prime radical | C31×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G | |
258 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 84 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)(34,38)(35,39,43,37)(36,40)(41,45,47,44)(42,46);(33,37)(35,43)(38,44)(39,40)(41,47)(45,46);(32,34,36,42)(33,38,40,46)(35,41,43,47)(37,44,39,45)806 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#7482, C31×#7486, C31×#7490, … |
| maximal normal subgroup of | C31×#7482, C31×#7486, C31×#7490, … |
| maximal quotient of | C31×#11431, C31×#11447, C31×#11476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 858);