Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23 |
| direct product | C22×ΓL(1,𝔽8)⋊C8 |
| derived, Sylow | C7 ⋊ C22×C3×C8 |
| Hall | C22×C7⋊C8 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C22×C8 |
152 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
112 conjugacy classes — view the full character table.
| maximal subgroup of | #89319⋊C3, #89324⋊C3, #89345⋊C3, … |
| maximal normal subgroup of | #89319⋊C3, #89324⋊C3, #89345⋊C3, … |
| maximal quotient of | C2×C4×ΓL(1,𝔽8)⋊C8, C2×C4⋊(ΓL(1,𝔽8)⋊C8), #89697⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 338);