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) = C22 |
| direct product | C2×#1136⋊C3 |
| unambiguous semidirect product | (C2×#1136)⋊C3 |
| Hall | C2×#1136 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×M16 |
| Sylow | C7 ⋊ C2×C3×M16 |
136 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
76 conjugacy classes — view the full character table.
| maximal subgroup of | #89318⋊C3, #89325⋊C3, #89327⋊C3, … |
| maximal normal subgroup of | #89318⋊C3, #89325⋊C3, #89327⋊C3, … |
| maximal quotient of | C2×#6377⋊C3, C4×#1136⋊C3, C2×C4⋊(ΓL(1,𝔽8)⋊C8), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 339);