Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×C7×M32 ⋊ C2 |
| Hall | C3×C7 ⋊ #504 |
| Sylow | C7 ⋊ #4907 |
| Sylow | C3 ⋊ #89679 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×D21 |
| commutator, socle | C2×C3×C7 | C22×C8 |
| Fitting | C3×C7×M32 | C2 |
| prime radical | #66728 | C4 |
| cosocle | C3×C7×C8 | C23 |
| hypercenter | M32 | D21 |
| hypocenter | C3×C7 | #504 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C8 ◁ M32 | |
168 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 672 |
Polycyclic presentation:
2^6,3,7|c^2=d,d^2=e,e^2=f,b^a=bf,g^a=g^2,h^a=h^6,c^b=cf240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 5833);