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×#3142 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C31 |
| prime radical | C31×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G | |
392 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 392 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) = 1984 |
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)(42,51)(43,46)(44,48)(45,50)(47,54)(49,52)(53,55);(33,36)(37,39)(40,42,52,53,45,54,44,43)(41,46,48,47,50,55,49,51);(32,34,35,38)(33,37,36,39)(40,43,44,54,45,53,52,42)(41,47,49,46,50,51,48,55);(40,44,45,52)(41,48,50,49)(42,53,54,43)(46,51,55,47)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#9442, C31×#10019, C31×#10021, … |
| maximal normal subgroup of | C31×#9442, C31×#10019, C31×#10021, … |
| maximal quotient of | C31×#20765, C31×#21093, C31×#21201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1874);