Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#1801 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C23×C31 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G | |
272 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 124 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)(36,37,39,43,40,42,48,50)(38,44,45,51,46,41,47,49);(32,33,34,35);(36,38,40,46)(37,41,42,44)(39,47,48,45)(43,51,50,49)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#7238, C31×#7323, C31×#8354, … |
| maximal normal subgroup of | C31×#7238, C31×#7323, C31×#8354, … |
| maximal quotient of | C31×#8558, C31×#8560, C31×#8562, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 533);