Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C32 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽8)×C32 |
| unambiguous semidirect product | (C7×C32)⋊C3 |
| Fitting, Hall | C7×C32 ⋊ C3 |
| derived, Sylow | C7 ⋊ C3×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C3×C32 |
| Frattini | C16 | C2×ΓL(1,𝔽8) |
| Fitting | C7×C32 | C3 |
| socle | C2×C7 | C3×C16 |
| prime radical | C2×ΓL(1,𝔽8) | C16 |
| cosocle | C7×C16 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C32 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)⋊C64, ΓL(1,𝔽8)×C64, C32×AGL(1,𝔽7), … |
| maximal normal subgroup of | ΓL(1,𝔽8)⋊C64, ΓL(1,𝔽8)×C64, C32×AGL(1,𝔽7), … |
| maximal quotient of | ΓL(1,𝔽8)×C64, C2×ΓL(1,𝔽8)×C32 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 2);