Finite Group DatabasebetaFinite Group Databasebeta| order | 8484 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C101 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽8)⋊C4×C101 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C4×C101) |
| unambiguous semidirect product | Dic7⋊(C3×C101) |
| unambiguous semidirect product | (Dic7×C101)⋊C3 |
| derived, Sylow | C7 ⋊ C3×C4×C101 |
| Hall | Dic7×C101 ⋊ C3 |
| Hall | ΓL(1,𝔽8)×C101 ⋊ C4 |
| Hall | C7×C101 ⋊ C3×C4 |
| Hall | Dic7 ⋊ C3×C101 |
| Hall | ΓL(1,𝔽8) ⋊ C4×C101 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C101 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C3×C4×C101 |
| Frattini | C2 | AGL(1,𝔽7)×C101 |
| Fitting, socle | C2×C7×C101 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C101 | |
24 conjugacy classes of subgroups.
1414 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×ΓL(1,𝔽8)⋊C4×C101, C101×C3⋊(ΓL(1,𝔽8)⋊C4) |
| maximal normal subgroup of | C3×ΓL(1,𝔽8)⋊C4×C101 |
| maximal quotient of | C101×Dic7⋊C9, C3×ΓL(1,𝔽8)⋊C4×C101, C101×C3⋊(ΓL(1,𝔽8)⋊C4) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8484, 11);