Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C42×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C3×C42) |
| Fitting | C42×C7 ⋊ C2×C3 |
| derived, Sylow | C7 ⋊ C2×C3×C42 |
| Hall | C42×D7 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C2×C3×C42 |
| Frattini | C22 | C22×AGL(1,𝔽7) |
| Fitting | C42×C7 | C2×C3 |
| socle, cosocle | C22×C7 | C23×C3 |
| prime radical | C22×AGL(1,𝔽7) | C22 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C42 | |
216 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
112 conjugacy classes — view the full character table.
| maximal subgroup of | C4×C8×AGL(1,𝔽7), #89457⋊C3, C4×#1131⋊C3, … |
| maximal normal subgroup of | C4×C8×AGL(1,𝔽7), #89457⋊C3, C4×#1131⋊C3, … |
| maximal quotient of | #89438⋊C3, #13051549, #89421⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 289);