Finite Group DatabasebetaFinite Group Databasebeta| order | 5607 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C9×C89 · Schur multiplier (H2) = C1 |
| direct product | C7⋊C9×C89 |
| unambiguous semidirect product | C7⋊(C9×C89) |
| unambiguous semidirect product | (C7×C89)⋊C9 |
| derived, Sylow | C7 ⋊ C9×C89 |
| Hall | C7×C89 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C89 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C9×C89 |
| Frattini | C3 | ΓL(1,𝔽8)×C89 |
| Fitting, socle | C3×C7×C89 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C89 | |
12 conjugacy classes of subgroups.
1335 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C7⋊C9×C89, C7⋊C9×D89, C89×D7⋊C9, … |
| maximal normal subgroup of | C2×C7⋊C9×C89, C7⋊C9×D89, C89×D7⋊C9, … |
| maximal quotient of | C2×C7⋊C9×C89, C89×C22⋊(C7⋊C9), C5×C7⋊C9×C89 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5607, 2);