Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C31 · Schur multiplier (H2) = C25 |
| direct product | C31×#3059 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C31 |
| prime radical | C2×#41×C31 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C2×#41×C31 ◁ G | |
460 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 200 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 — 4 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)(32,33)(34,35)(36,37,41,45)(38,46,50,60)(39,43,51,57)(40,53,47,44)(42,54,56,49)(48,59,62,67)(52,64,61,58)(55,65,66,63);(36,38,47,54,41,50,40,49)(37,42,53,60,45,56,44,46)(39,48,61,65,51,62,52,63)(43,55,64,67,57,66,58,59);(33,35)(36,39)(37,43)(38,48)(40,52)(41,51)(42,55)(44,58)(45,57)(46,59)(47,61)(49,63)(50,62)(53,64)(54,65)(56,66)(60,67);(39,51)(43,57)(48,62)(52,61)(55,66)(58,64)(59,67)(63,65)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#11869, C31×#11871, C31×#13075, … |
| maximal normal subgroup of | C31×#11869, C31×#11871, C31×#13075, … |
| maximal quotient of | C31×#19188, C31×#19193, C31×#19199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1791);