Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#3261 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C31 |
| prime radical | C23×C31 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C31×#117 ◁ G | |
322 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 1984 |
A minimal faithful permutation representation — 4 generators on 71 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)(33,36)(34,38)(40,41,45,49)(42,46,51,60)(43,53,55,63)(44,56,58,48)(47,57,62,54)(50,64,65,71)(52,68,67,61)(59,69,70,66);(32,33)(34,37)(35,36)(38,39)(40,42,45,51)(41,46,49,60)(43,50,55,65)(44,52,58,67)(47,59,62,70)(48,61,56,68)(53,64,63,71)(54,66,57,69);(33,36)(37,39)(40,43,45,55)(41,47,49,62)(42,50,51,65)(44,57,58,54)(46,59,60,70)(48,63,56,53)(52,69,67,66)(61,71,68,64);(32,34,35,38)(33,37,36,39)(42,51)(46,60)(50,65)(52,67)(59,70)(61,68)(64,71)(66,69)806 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12165, C31×#12168, C31×#12171, … |
| maximal normal subgroup of | C31×#12165, C31×#12168, C31×#12171, … |
| maximal quotient of | C31×#21898, C31×#21968, C31×#22124, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1993);