Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#2048 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C2×C31×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G | |
274 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 496 |
A minimal faithful permutation representation — 3 generators on 47 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)(33,38)(34,41)(35,36)(39,46)(40,44)(42,43);(32,33)(34,39)(35,44)(36,38)(37,40)(41,45)(42,47)(43,46);(32,34,35,42,37,43,36,41)(33,39,38,45,40,46,44,47)713 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C31×#2048, C31×#23525, C31×#24295, … |
| maximal normal subgroup of | C2×C31×#2048, C31×#23525, C31×#24295, … |
| maximal quotient of | C31×#10440, C31×#10441, C31×#10442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 780);