Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#2078 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C2×#41×C31 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C22×C4×C31 ◁ G | |
198 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 3 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)(40,41,43,48,45,49,44,47)(42,46,51,54,52,53,50,55);(33,36)(37,39)(40,42)(41,46)(43,51)(44,50)(45,52)(47,55)(48,54)(49,53);(32,34,35,38)(33,37,36,39)(41,47)(42,50)(43,44)(46,53)(48,49)(51,52)806 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12284, C31×#12285, C31×#12286, … |
| maximal normal subgroup of | C31×#12284, C31×#12285, C31×#12286, … |
| maximal quotient of | C31×#10615, C31×#10616, C31×#10617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 810);