Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C31 · Schur multiplier (H2) = C23×C4 |
| direct product | C31×#3140 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C31 |
| prime radical | C22×D4×C31 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G | |
400 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 4 generators on 55 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)(42,50)(43,46)(44,54)(45,49)(47,52)(48,51)(53,55);(33,36)(37,39)(40,42,51,53,45,52,44,43)(41,46,54,47,49,55,48,50);(32,34,35,38)(33,37,36,39)(40,43,44,52,45,53,51,42)(41,47,48,46,49,50,54,55);(42,52)(43,53)(46,55)(47,50)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#9439, C31×#10022, C31×#10024, … |
| maximal normal subgroup of | C31×#9439, C31×#10022, C31×#10024, … |
| maximal quotient of | C31×#20765, C31×#21094, C31×#21095, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1872);