Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C25 |
| direct product | C31×#1791 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C31 | #120 |
| commutator | C23 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C2×C31 | #379 |
| prime radical | C31×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C31 ◁ C22×C4×C31 ◁ G | |
548 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 992 |
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)(32,33,36,41,37,40,49,54)(34,42,45,55,44,38,48,51)(35,39,47,52,46,53,59,63)(43,50,58,61,57,56,60,62);(32,34)(33,38)(35,43)(36,48)(37,44)(39,50)(40,42)(41,55)(45,49)(46,57)(47,60)(51,54)(52,62)(53,56)(58,59)(61,63);(32,35)(33,39)(34,43)(36,47)(37,46)(38,50)(40,53)(41,52)(42,56)(44,57)(45,58)(48,60)(49,59)(51,61)(54,63)(55,62)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#13045, C31×#13046, C31×#13051, … |
| maximal normal subgroup of | C31×#13045, C31×#13046, C31×#13051, … |
| maximal quotient of | C31×#8316, C31×#8317, C31×#8348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 523);