Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#1497 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C2×D4 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C42 | C23×C31 |
| socle | C22×C31 | C2×C22⋊C4 |
| prime radical | C23×C31 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G | |
202 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 1984 |
A minimal faithful permutation representation — 3 generators on 63 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)(34,41)(35,39)(36,45)(37,40)(38,43)(42,46)(44,47)(48,49,51,55,52,54,58,61)(50,53,57,59,56,60,62,63);(32,34,37,44)(33,38,40,46)(35,43,36,42)(39,47,45,41)(48,50)(49,53)(51,57)(52,56)(54,60)(55,59)(58,62)(61,63);(32,35,37,36)(33,39,40,45)(34,42,44,43)(38,41,46,47)(49,54)(50,56)(55,61)(57,62)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#8490, C31×#8501, C31×#8502, … |
| maximal normal subgroup of | C31×#8490, C31×#8501, C31×#8502, … |
| maximal quotient of | C31×#7701, C31×#7702, C31×#7703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 229);