Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22×C4 |
| direct product | C31×#1983 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C31 | C2×D4 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C22×D4×C31 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G | |
344 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 128 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 — 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)(33,35)(37,39)(40,41,44,45,43,46,42,47);(32,33,34,35)(41,45)(42,44)(46,47);(36,37,38,39)(41,46)(45,47)1178 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#8510, C31×#8511, C31×#10508, … |
| maximal normal subgroup of | C31×#8510, C31×#8511, C31×#10508, … |
| maximal quotient of | C31×#9675, C31×#9676, C31×#9697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 715);