Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#1946 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C2×D4 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C22×C31 | C4×D4 |
| prime radical | C23×C31 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C22×C8×C31 ◁ G | |
200 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 112 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)(36,37,40,45,41,46,50,44)(38,42,39,48,47,43,49,51);(33,35)(36,38,40,39,41,47,50,49)(37,42,45,48,46,43,44,51);(36,39,40,47,41,49,50,38)(37,43,44,48,46,42,45,51)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#7125, C31×#12944, C31×#12956, … |
| maximal normal subgroup of | C31×#7125, C31×#12944, C31×#12956, … |
| maximal quotient of | C31×#9595, C31×#9596, C31×#9597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 678);