Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×C7×Q8 ⋊ C2×C3 |
| Hall | C22×#1161 ⋊ C3 |
| Sylow | C7 ⋊ A4×#41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D7⋊A4 |
| commutator, socle, cosocle | C23×C7 | C23×C3 |
| Fitting | C22×C7×Q8 | C2×C3 |
| hypercenter | Q8 | D7⋊A4 |
| hypocenter | C22×C7 | C3×#41 |
| LCS | G ▷ C23×C7 ▷ C22×C7 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
430 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2^3,3,2^3,7|b^2=e,c^2=e,c^a=ce,h^a=h^6,c^b=ce,f^d=fg,g^d=f,h^d=h^460 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11479);