Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C31 · Schur multiplier (H2) = C26 |
| direct product | C31×#2888 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | C22×C22⋊C4 |
| commutator | C22 | C23×C4×C31 |
| Frattini, cosocle | C2×C4 | C24×C31 |
| prime radical | C2×C31×H(5,𝔽2) | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C2×#41×C31 ◁ G | |
756 conjugacy classes of subgroups; only the 340 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 340 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 — 4 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)(32,33,35,39,36,37,42,44)(34,38,41,45,40,43,46,47);(33,37)(35,42)(38,43)(41,46);(32,34)(33,38)(35,41)(36,40)(37,43)(39,45)(42,46)(44,47);(34,40)(38,43)(41,46)(45,47)1271 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#13097, C31×#13099, C31×#21778, … |
| maximal normal subgroup of | C31×#13097, C31×#13099, C31×#21778, … |
| maximal quotient of | C31×#17350, C31×#17353, C31×#17367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1620);