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×#1979 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | #121 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C23×C31 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C22×C4×C31 ◁ G | |
276 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 276 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) = 67 |
| transitive degree | μt(G) = 1984 |
A minimal faithful permutation representation — 3 generators on 67 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)(36,37,41,46,42,45,40,54)(38,43,49,58,48,59,55,57)(39,50,52,60,53,64,51,44)(47,61,63,56,62,66,65,67);(32,33,34,35)(36,38,40,49,42,48,41,55)(37,43,46,57,45,59,54,58)(39,47,51,63,53,62,52,65)(44,56,50,61,60,67,64,66);(36,39,41,52,42,53,40,51)(37,44,54,64,45,60,46,50)(38,47,55,65,48,62,49,63)(43,56,58,66,59,67,57,61)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#13019, C31×#13020, C31×#13023, … |
| maximal normal subgroup of | C31×#13019, C31×#13020, C31×#13023, … |
| maximal quotient of | C31×#9669, C31×#9671, C31×#9725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 711);