Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #6445⋊C3 |
| Fitting | C7×C22⋊C4 ⋊ C2×C3 |
| Hall | #6445 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #120 |
| Sylow | C7 ⋊ C3×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C23×C3 |
| Fitting | C7×C22⋊C4 | C2×C3 |
| hypercenter | C22⋊C4 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#120 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
260 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 56 |
Polycyclic presentation:
2^3,3,2^2,7|c^2=e,c^a=ce,g^a=g^6,c^b=cf,g^d=g^452 conjugacy classes — view the full character table.
| maximal subgroup of | #89513⋊C3, C2×#6445⋊C3, #90181⋊C3, … |
| maximal normal subgroup of | #89513⋊C3, C2×#6445⋊C3, #90181⋊C3, … |
| maximal quotient of | #89428⋊C3, #89445⋊C3, #89437⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 300);