Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#2221 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C23×C31 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C31×#131 ◁ G | |
148 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 60 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)(33,39)(34,45)(35,40)(36,37)(41,56)(42,53)(43,52)(44,54)(46,60)(47,50)(48,49)(51,57)(58,63)(59,62);(32,33,40,53,37,43,49,54,38,44,48,52,36,42,35,39)(34,41,55,56,45,57,60,62,47,58,61,63,50,59,46,51);(32,34)(33,41)(35,46)(36,50)(37,45)(38,47)(39,51)(40,55)(42,59)(43,57)(44,58)(48,61)(49,60)(52,63)(53,56)(54,62)806 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#13669, C31×#13670, C31×#13700, … |
| maximal normal subgroup of | C31×#13669, C31×#13670, C31×#13700, … |
| maximal quotient of | C31×#12430, C31×#12433, C31×#12439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 953);