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) = C23 |
| direct product | C4×C4⋊AGL(1,𝔽7) |
| unambiguous semidirect product | (C4×D28)⋊C3 |
| Fitting | C42×C7 ⋊ C2×C3 |
| Hall | C4×D28 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C4×D4 |
| Sylow | C7 ⋊ C3×C4×D4 |
188 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 77 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 | #89254⋊C3, #89278⋊C3, C8×C4⋊AGL(1,𝔽7), … |
| maximal normal subgroup of | #89254⋊C3, #89278⋊C3, C8×C4⋊AGL(1,𝔽7), … |
| maximal quotient of | #89425⋊C3, #89437⋊C3, #89440⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 291);