Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#2127 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | C2×#99 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C2×D4 | C23×C31 |
| prime radical | C31×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C23×C31 ◁ C22×D4×C31 ◁ G | |
286 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 286 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)(34,40,41,47)(35,43,45,38)(36,39)(37,42)(44,46);(32,34)(33,37)(35,44)(36,41)(38,47)(39,46)(40,43)(42,45);(34,41)(37,46)(40,47)(42,44)713 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12707, C31×#12708, C31×#12727, … |
| maximal normal subgroup of | C31×#12707, C31×#12708, C31×#12727, … |
| maximal quotient of | C31×#11432, C31×#11448, C31×#11479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 859);