Finite Group DatabasebetaFinite Group Databasebeta| order | 17178 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C409 · Schur multiplier (H2) = C1 |
| direct product | AGL(1,𝔽7)×C409 |
| unambiguous semidirect product | D7⋊(C3×C409) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C2×C409) |
| Fitting, socle, Hall | C7×C409 ⋊ C2×C3 |
| derived, Sylow | C7 ⋊ C2×C3×C409 |
| Hall | ΓL(1,𝔽8)×C409 ⋊ C2 |
| Hall | D7×C409 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×C409 |
| Hall | D7 ⋊ C3×C409 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C409 | AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C7 | C2×C3×C409 |
| Frattini | C1 | G |
| Fitting, socle | C7×C409 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C409 | |
16 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2863 |
2863 conjugacy classes. The character table is too large to display.
| maximal subgroup of | AGL(1,𝔽7)×D409, C2×AGL(1,𝔽7)×C409 |
| maximal normal subgroup of | AGL(1,𝔽7)×D409, C2×AGL(1,𝔽7)×C409 |
| maximal quotient of | ΓL(1,𝔽8)⋊C4×C409, C2×AGL(1,𝔽7)×C409 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17178, 18);