Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#2009 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C31×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G | |
348 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 88 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)(34,44)(35,48)(36,42)(37,41)(38,43)(39,46)(40,50)(45,53)(47,58)(49,57)(51,56)(52,54)(55,61)(59,60)(62,63);(32,34)(33,39)(35,45)(36,52)(37,46)(38,47)(40,55)(41,58)(42,44)(43,54)(48,59)(49,62)(50,60)(51,61)(53,57)(56,63);(32,35,38,51)(33,40,43,57)(34,45,47,61)(36,50,37,49)(39,53,54,55)(41,48,42,56)(44,59,58,63)(46,62,52,60)713 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C31×#2009, C31×#23412, C31×#23873, … |
| maximal normal subgroup of | C2×C31×#2009, C31×#23412, C31×#23873, … |
| maximal quotient of | C31×#9931, C31×#9933, C31×#9935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 741);