Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Hall | C3×C7 ⋊ D4×Q8 |
| Sylow | C7 ⋊ #5727 |
| Sylow | C3 ⋊ Q8×D28 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3×D7 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C42×C7 | C22 |
| prime radical | C2×D84 | C22 |
| hypercenter | C42 | D3×D7 |
| hypocenter | C3×C7 | D4×Q8 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C42 | |
560 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 160 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|a^2=f,b^2=f,c^2=f,d^2=e,c^a=cf,d^a=de,h^a=h^6,c^b=cf,g^b=g^2147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 7038);