Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊Q8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×(C22⋊Dic7)⋊C3 |
| unambiguous semidirect product | (C4×C22⋊Dic7)⋊C3 |
| Hall | C4×C22⋊Dic7 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C4×C22⋊C4 |
| Sylow | C7 ⋊ C3×C4×C22⋊C4 |
516 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
152 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×(C22⋊Dic7)⋊C3, C4×C22⋊C4×AGL(1,𝔽7), C4×#10620789, … |
| maximal normal subgroup of | C4×C22⋊C4×AGL(1,𝔽7), C4×#10620789, C4×#6500⋊C3, … |
| maximal quotient of | AGL(1,𝔽5)×(C22⋊Dic7)⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1398);