Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | D7⋊#994 |
| Fitting | C7×H(5,𝔽2) ⋊ C2×C3 |
| Hall | D7×H(5,𝔽2) ⋊ C3 |
| Sylow | C7 ⋊ C2×#994 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D7⋊A4 |
| commutator, cosocle, hypocenter | C7×Q8 | C23×C3 |
| Fitting | C7×H(5,𝔽2) | C2×C3 |
| socle | C2×C7 | C23×A4 |
| hypercenter | Q8 | D7⋊A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
416 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 62 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:
2^3,3,2^2,7,2|b^2=h,c^2=h,e^2=h,f^2=h,g^a=g^6,c^b=ch,e^d=ef,f^d=e,g^d=g^4,f^e=fh55 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#994, #72246243, #72246245 |
| maximal normal subgroup of | AGL(1,𝔽7)×#994, #72246243, #72246245 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11486);