Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#1389 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C31 | #102 |
| commutator | C8 | C42×C31 |
| Frattini, cosocle | C22×C8 | C22×C31 |
| socle | C2×C31 | #340 |
| prime radical | C23×C31 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C31 ◁ C22×C4×C31 ◁ C22×C8×C31 ◁ G | |
128 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 2 generators on 63 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)(32,33,36,41,38,42,37,53)(34,43,45,60,48,56,47,62)(35,49,50,58,52,63,51,40)(39,54,55,46,57,61,59,44);(32,34,35,46,36,45,50,61,38,48,52,44,37,47,51,54)(33,39,40,56,53,59,63,60,42,57,58,43,41,55,49,62)992 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C31×#1389, C31×#12646, C31×#12649, … |
| maximal normal subgroup of | C2×C31×#1389, C31×#12646, C31×#12649, … |
| maximal quotient of | C31×#7159, C31×#7175, C31×#7177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 121);