Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#1909 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #386 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C2×#41×C31 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G | |
312 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 108 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)(34,42)(35,46,48,39)(36,49)(37,41)(38,44)(40,51)(43,56,59,63)(45,57)(47,54,50,55)(52,58,62,60)(53,61);(32,34,36,44,37,45,51,61)(33,38,40,42,41,53,49,57)(35,43,50,60,48,59,47,58)(39,52,55,56,46,62,54,63);(32,35)(33,39)(34,43)(36,50)(37,48)(38,52)(40,55)(41,46)(42,56)(44,60)(45,59)(47,51)(49,54)(53,62)(57,63)(58,61)899 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12756, C31×#12757, C31×#12779, … |
| maximal normal subgroup of | C31×#12756, C31×#12757, C31×#12779, … |
| maximal quotient of | C31×#9099, C31×#9100, C31×#9103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 641);