Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #66665 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C4×D7⋊SL(2,𝔽3) |
| unambiguous semidirect product | D7⋊(C4×SL(2,𝔽3)) |
| unambiguous semidirect product | (C4×D7)⋊SL(2,𝔽3) |
| Fitting | C4×C7×Q8 ⋊ C2×C3 |
| derived | C7×Q8 ⋊ C2×C3×C4 |
| Hall | C4×Q8×D7 ⋊ C3 |
| Sylow | C7 ⋊ C2×C4×SL(2,𝔽3) |
216 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
76 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×D7⋊SL(2,𝔽3), C4×SL(2,𝔽3)×AGL(1,𝔽7) |
| maximal normal subgroup of | C4×SL(2,𝔽3)×AGL(1,𝔽7) |
| maximal quotient of | AGL(1,𝔽5)×D7⋊SL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6929);