Finite Group DatabasebetaFinite Group Databasebeta| order | 8442 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| Fitting, socle, derived, Hall | C7×C67 ⋊ C2×C32 |
| Hall | ΓL(1,𝔽8)×C67⋊C3 ⋊ C2 |
| Hall | D469 ⋊ C32 |
| Sylow | C67 ⋊ C3×AGL(1,𝔽7) |
| Sylow | C7 ⋊ C3×D67⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C7×C67 | C2×C32 |
| LCS | G ▷ C7×C67 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
Polycyclic presentation:
2,3^2,7,67|d^a=d^6,e^a=e^66,e^b=e^29,d^c=d^276 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×D67⋊C3, C2×ΓL(1,𝔽8)⋊(D67⋊C3) |
| maximal normal subgroup of | AGL(1,𝔽7)×D67⋊C3, C2×ΓL(1,𝔽8)⋊(D67⋊C3) |
| maximal quotient of | ΓL(1,𝔽8)⋊((C67⋊C3)⋊C4), C2×ΓL(1,𝔽8)⋊(D67⋊C3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8442, 26);