Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊Q8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×(C2×AGL(1,𝔽7))⋊C4 |
| unambiguous semidirect product | (C2×(C2×D7)⋊C4)⋊C3 |
| Fitting | C22×C4×C7 ⋊ C2×C3 |
| Hall | C2×(C2×D7)⋊C4 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×C22⋊C4 |
| Sylow | C7 ⋊ C2×C3×C22⋊C4 |
264 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
76 conjugacy classes — view the full character table.
| maximal subgroup of | #89263⋊C3, #89438⋊C3, #89439⋊C3, … |
| maximal normal subgroup of | #89263⋊C3, #89438⋊C3, #89439⋊C3, … |
| maximal quotient of | #89701⋊C3, C2×#6379⋊C3, C4×(C2×AGL(1,𝔽7))⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 345);