Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C31 ⋊ #2399 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D31 |
| commutator, socle, prime radical, cosocle | C23×C31 | C24 |
| Frattini | C23 | C23×D31 |
| Fitting | C31×#376 | C2 |
| hypercenter | #376 | D31 |
| hypocenter | C31 | #2399 |
| LCS | G ▷ C23×C31 ▷ C31 | |
| UCS | C1 ◁ C22×C4 ◁ #376 | |
504 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 189 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=e,b^2=e,c^2=e,d^2=ef,b^a=be,c^a=cf,c^b=cg,h^b=h^30,h^c=h^30644 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 7803);