Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×C4×ΓL(1,𝔽8)⋊C4 |
| unambiguous semidirect product | (C2×ΓL(1,𝔽8))⋊C42 |
| unambiguous semidirect product | (C2×C4×Dic7)⋊C3 |
| derived, Sylow | C7 ⋊ C2×C3×C42 |
| Hall | C2×C4×Dic7 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×C42 |
216 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
112 conjugacy classes — view the full character table.
| maximal subgroup of | #89327⋊C3, #89345⋊C3, #89350⋊C3, … |
| maximal normal subgroup of | #89327⋊C3, #89345⋊C3, #89350⋊C3, … |
| maximal quotient of | C2×C4×ΓL(1,𝔽8)⋊C8, C2×#6377⋊C3, C4×#1136⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 340);