Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22×C4 |
| direct product | C31×#2122 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | C2×#99 |
| commutator | C23 | C22×C4×C31 |
| Frattini, cosocle | C2×D4 | C23×C31 |
| prime radical | C31×#535 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G | |
266 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 266 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)(34,38)(35,42,43,37)(36,39)(40,46)(41,45,47,44);(33,37)(34,40)(38,45)(39,42)(41,47)(44,46);(32,34,36,40)(33,38,39,46)(35,41,43,47)(37,44,42,45)713 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12715, C31×#12718, C31×#12721, … |
| maximal normal subgroup of | C31×#12715, C31×#12718, C31×#12721, … |
| maximal quotient of | C31×#11427, C31×#11454, C31×#11473, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 854);