Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | Q8×ΓL(1,𝔽8)⋊C4 |
| Hall | Q8×Dic7 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C4×Q8 |
| Sylow | C7 ⋊ C3×C4×Q8 |
140 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | #89563⋊C3, #89566⋊C3, #89568⋊C3, … |
| maximal normal subgroup of | #89563⋊C3, #89566⋊C3, #89568⋊C3, … |
| maximal quotient of | C4⋊C4×ΓL(1,𝔽8)⋊C4, #89753⋊C3, #89757⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 363);