Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C2 |
| direct product | C31×#2237 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C31 |
| prime radical | C23×C31 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C22×C8×C31 ◁ G | |
132 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 132 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) = 71 |
| transitive degree | μt(G) = 1984 |
A minimal faithful permutation representation — 3 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)(32,33)(34,37)(35,36)(38,39)(40,41,45,50)(42,46,53,61)(43,47,56,63)(44,57,55,49)(48,64,62,54)(51,58,67,70)(52,68,66,60)(59,71,69,65);(33,36)(37,39)(40,42,45,53)(41,46,50,61)(43,51,56,67)(44,52,55,66)(47,58,63,70)(48,59,62,69)(49,60,57,68)(54,65,64,71);(32,34,35,38)(33,37,36,39)(41,47)(43,54)(44,55)(46,58)(48,49)(50,63)(51,65)(52,66)(56,64)(57,62)(59,60)(61,70)(67,71)(68,69)899 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C31×#2237, C31×#33645, C31×#33647, … |
| maximal normal subgroup of | C2×C31×#2237, C31×#33645, C31×#33647, … |
| maximal quotient of | C31×#12868, C31×#12921, C31×#12994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 969);