Finite Group DatabasebetaFinite Group Databasebeta| order | 15603 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C743 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽8)×C743 |
| unambiguous semidirect product | C7⋊(C3×C743) |
| unambiguous semidirect product | (C7×C743)⋊C3 |
| Fitting, socle, Hall | C7×C743 ⋊ C3 |
| derived, Sylow | C7 ⋊ C3×C743 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C743 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C3×C743 |
| Frattini | C1 | G |
| Fitting, socle | C7×C743 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C743 | |
8 conjugacy classes of subgroups.
| transitive degree | μt(G) = 5201 |
3715 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×ΓL(1,𝔽8)×C743, ΓL(1,𝔽8)×D743, AGL(1,𝔽7)×C743, … |
| maximal normal subgroup of | C2×ΓL(1,𝔽8)×C743, ΓL(1,𝔽8)×D743, AGL(1,𝔽7)×C743, … |
| maximal quotient of | C2×ΓL(1,𝔽8)×C743 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15603, 2);