Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#1817 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C31 | C23 |
| commutator | C22 | C22×C8×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C23×C31 | C22×C4 |
| prime radical | C24×C31 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C31 ◁ G | |
380 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1984 |
A minimal faithful permutation representation — 3 generators on 51 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)(38,41)(39,40)(42,43)(44,45,46,48,47,49,50,51);(37,40)(41,43);(33,35)(36,38,39,42)(37,41,40,43)1736 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#7017, C31×#7021, C31×#7033, … |
| maximal normal subgroup of | C31×#7017, C31×#7021, C31×#7033, … |
| maximal quotient of | C31×#8698, C31×#8699, C31×#8700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 549);