Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C22⋊(C7⋊C8))⋊C3 |
| Hall | C22⋊(C7⋊C8) ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C22⋊C8 |
| Sylow | C7 ⋊ C3×C22⋊C8 |
100 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^2,3,2^3,7|a^2=d,d^2=f,b^a=be,g^a=g^6,g^c=g^476 conjugacy classes — view the full character table.
| maximal subgroup of | #89246⋊C3, #89247⋊C3, #89263⋊C3, … |
| maximal normal subgroup of | #89246⋊C3, #89247⋊C3, #89263⋊C3, … |
| maximal quotient of | #89319⋊C3, #89320⋊C3, #89323⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 40);