Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C2 |
| direct product | C31×#1676 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C31 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C31 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C42×C31 ◁ G | |
144 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 3968 |
A minimal faithful permutation representation — 3 generators on 63 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)(32,33,36,39)(34,40,43,47)(35,44,41,38)(37,45,46,42)(48,49,52,55)(50,56,58,63)(51,59,61,54)(53,60,62,57);(32,34,41,45,36,43,35,42)(33,37,44,47,39,46,38,40);(48,50,52,58)(49,53,55,62)(51,60,61,57)(54,63,59,56)899 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12384, C31×#12385, C31×#12386, … |
| maximal normal subgroup of | C31×#12384, C31×#12385, C31×#12386, … |
| maximal quotient of | C31×#8645, C31×#9216, C31×#9568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 408);