Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of D7⋊(#120⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | D7⋊(#120⋊C3) |
| unambiguous semidirect product | (D7⋊(C22⋊A4))⋊C2 |
| Fitting | C7×#120 ⋊ C2×C3 |
| derived | C24×C7 ⋊ C22×C3 |
| Hall | D7×#120 ⋊ C3 |
| Sylow | C7 ⋊ C2×#120⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C24×C7 | C22×C3 |
| Frattini | C22 | C2×D7⋊A4 |
| Fitting | C7×#120 | C2×C3 |
| socle | C22×C7 | C22×A4 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 | |
284 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 56 |
Polycyclic presentation:
2^2,3,2^4,7|h^a=h^6,d^b=df,e^b=efg,d^c=de,e^c=d,f^c=g,g^c=fg,h^c=h^234 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#120⋊C3, #72215058, C2×D7⋊(#120⋊C3), … |
| maximal normal subgroup of | AGL(1,𝔽7)×#120⋊C3, C2×D7⋊(#120⋊C3) |
| maximal quotient of | #72215058, #72276711, C2×D7⋊(#120⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6933);