Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C31 ⋊ #1840 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D31 |
| commutator, socle, prime radical | C23×C31 | C22×C4 |
| Fitting | C31×#336 | C2 |
| cosocle | C22×C4×C31 | C23 |
| hypercenter | #336 | D31 |
| hypocenter | C31 | #1840 |
| LCS | G ▷ C23×C31 ▷ C31 | |
| UCS | C1 ◁ C22×C4 ◁ #336 | |
196 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 3968 |
Polycyclic presentation:
2^7,31|a^2=f,b^2=de,c^2=d,f^2=g,b^a=bd,c^a=ce,c^b=cg,h^c=h^30644 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 4654);