Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Sylow | C31 ⋊ #2862 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D31 |
| commutator, socle, prime radical, cosocle | C23×C31 | C24 |
| Fitting | C2×C31×C4⋊Q8 | C2 |
| hypercenter | C2×C4⋊Q8 | D31 |
| hypocenter | C31 | #2862 |
| LCS | G ▷ C23×C31 ▷ C31 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
504 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 197 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=g,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dg,h^b=h^30,h^d=h^30464 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 12887);