Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#1738 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C2×D4 |
| commutator | C4 | C2×C42×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C22×C31 | C2×C4⋊C4 |
| prime radical | C24×C31 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C23×C4×C31 ◁ G | |
372 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 992 |
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)(36,37,39,44,41,45,40,43)(38,46,48,42,47,51,49,50);(33,35)(36,38,40,49,41,47,39,48)(37,42,44,51,45,50,43,46);(38,47)(42,50)(46,51)(48,49)1364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#10744, C31×#10753, C31×#10767, … |
| maximal normal subgroup of | C31×#10744, C31×#10753, C31×#10767, … |
| maximal quotient of | C31×#7873, C31×#7876, C31×#7877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 470);