Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#2080 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C31 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C2×C31 | #397 |
| prime radical | C2×#41×C31 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C31 ◁ C22×C4×C31 ◁ G | |
190 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 72 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)(34,42,45,55)(35,39)(36,40)(37,41)(38,44,52,49)(43,58,46,57)(47,53)(48,54)(50,56)(51,60,62,59)(61,63);(32,34)(33,38)(35,46)(36,49)(37,45)(39,51)(40,55)(41,52)(42,56)(43,48)(44,50)(47,59)(53,58)(54,62)(57,63)(60,61);(32,35,36,47,37,48,50,61)(33,39,40,53,41,54,56,63)(34,43,44,59,45,46,49,60)(38,51,42,57,52,62,55,58)806 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C31×#2080, C31×#23649, C31×#23833, … |
| maximal normal subgroup of | C2×C31×#2080, C31×#23649, C31×#23833, … |
| maximal quotient of | C31×#10627, C31×#10628, C31×#10629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 812);