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 | C2×(ΓL(1,𝔽8)⋊C4)⋊C4 |
| unambiguous semidirect product | (C2×Dic7⋊C4)⋊C3 |
| Hall | C2×Dic7⋊C4 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×C4⋊C4 |
| Sylow | C7 ⋊ C2×C3×C4⋊C4 |
184 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
76 conjugacy classes — view the full character table.
| maximal subgroup of | #89264⋊C3, #89439⋊C3, #89418⋊C3, … |
| maximal normal subgroup of | #89264⋊C3, #89439⋊C3, #89418⋊C3, … |
| maximal quotient of | C4×(ΓL(1,𝔽8)⋊C4)⋊C4, #89700⋊C3, #89714⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 341);