Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#2047 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C23×C31 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C22×C4×C31 ◁ G | |
256 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 84 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)(32,33)(34,35)(36,37,41,45)(38,46,49,61)(39,50,53,43)(40,54,56,44)(42,55,58,48)(47,63,51,62)(52,60,67,65)(57,64,59,66);(36,38,41,49)(37,42,45,58)(39,51,53,47)(40,55,56,48)(43,59,50,57)(44,61,54,46)(52,64,67,66)(60,62,65,63);(33,35)(36,39,40,52,41,53,56,67)(37,43,44,60,45,50,54,65)(38,47,48,64,49,51,55,66)(42,57,46,62,58,59,61,63)806 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12167, C31×#12168, C31×#12169, … |
| maximal normal subgroup of | C31×#12167, C31×#12168, C31×#12169, … |
| maximal quotient of | C31×#10446, C31×#10447, C31×#10454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 779);