Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#2049 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C2×#41×C31 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G | |
254 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 3 generators on 63 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)(32,33,38,43)(34,44,47,59)(35,48,52,57)(36,42,37,41)(39,53,55,46)(40,50,58,51)(45,56,49,54)(60,63,62,61);(32,34)(33,39)(35,49)(36,53)(37,46)(38,47)(40,56)(41,59)(42,44)(43,55)(45,52)(48,62)(50,61)(51,63)(54,58)(57,60);(32,35,36,50,38,52,37,51)(33,40,41,57,43,58,42,48)(34,45,46,61,47,49,53,63)(39,54,44,60,55,56,59,62)713 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C31×#2049, C31×#23525, C31×#24295, … |
| maximal normal subgroup of | C2×C31×#2049, C31×#23525, C31×#24295, … |
| maximal quotient of | C31×#10444, C31×#10445, C31×#10446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 781);