Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽8)×Q32 |
| unambiguous semidirect product | (C7×Q32)⋊C3 |
| Fitting, Hall | C7×Q32 ⋊ C3 |
| Sylow | C7 ⋊ C3×Q32 |
48 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
55 conjugacy classes — view the full character table.
| maximal subgroup of | #89316⋊C3, #89317⋊C3, ΓL(1,𝔽8)×SD64, … |
| maximal normal subgroup of | #89316⋊C3, #89317⋊C3, ΓL(1,𝔽8)×SD64, … |
| maximal quotient of | ΓL(1,𝔽8)×#358, ΓL(1,𝔽8)×#366, C2×ΓL(1,𝔽8)×Q32 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 133);