Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#1841 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C31 | #117 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C2×C31 | #383 |
| prime radical | C23×C31 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C31 ◁ C2×M16×C31 ◁ G | |
162 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 496 |
A minimal faithful permutation representation — 3 generators on 47 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)(34,42,45,46)(35,38)(36,40)(37,41)(39,44,47,43);(33,38,41,40)(39,46,47,42);(32,34,35,43,37,45,36,44)(33,39,40,42,41,47,38,46)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#7225, C31×#7226, C31×#11836, … |
| maximal normal subgroup of | C31×#7225, C31×#7226, C31×#11836, … |
| maximal quotient of | C31×#8800, C31×#8801, C31×#8802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 573);