Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#2971 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C24 |
| commutator | C22 | C23×C4×C31 |
| Frattini, cosocle | C2×C4 | C24×C31 |
| socle | C22×C31 | C23×C4 |
| prime radical | C2×#41×C31 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ G | |
402 conjugacy classes of subgroups; only the 276 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 276 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) = 1984 |
A minimal faithful permutation representation — 4 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)(32,33)(34,37)(35,36)(38,39)(40,41,45,42,44,46,43,47);(33,36)(37,39)(41,46)(42,47);(40,42,45,46,44,47,43,41);(32,34,35,38)(33,37,36,39)(40,43,44,45)(41,42,46,47)1550 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#9160, C31×#9162, C31×#9350, … |
| maximal normal subgroup of | C31×#9160, C31×#9162, C31×#9350, … |
| maximal quotient of | C31×#20480, C31×#20494, Q8×C31×#97, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1703);