Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22×C4 |
| direct product | C31×#1965 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C2×C31×#127 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G | |
396 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 3 generators on 43 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)(36,37,39,42,40,41,43,38);(36,38,39,41,40,42,43,37);(32,33,34,35)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#8469, C31×#8472, C31×#9253, … |
| maximal normal subgroup of | C31×#8469, C31×#8472, C31×#9253, … |
| maximal quotient of | C31×#9649, C31×#9653, C31×#9685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 697);