Finite Group DatabasebetaFinite Group Databasebeta| order | 17451 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C277 · Schur multiplier (H2) = C3 |
| direct product | C3×ΓL(1,𝔽8)×C277 |
| unambiguous semidirect product | C7⋊(C32×C277) |
| unambiguous semidirect product | (C3×C7)⋊(C3×C277) |
| Fitting, socle | C3×C7×C277 ⋊ C3 |
| derived, Sylow | C7 ⋊ C32×C277 |
| Hall | C7×C277 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C277 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C32×C277 |
| Frattini | C1 | G |
| Fitting, socle | C3×C7×C277 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C277 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 5817 |
4155 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×AGL(1,𝔽7)×C277, C3×ΓL(1,𝔽8)⋊D277, C3×ΓL(1,𝔽8)×D277, … |
| maximal normal subgroup of | C3×AGL(1,𝔽7)×C277, C3×ΓL(1,𝔽8)⋊D277, C3×ΓL(1,𝔽8)×D277, … |
| maximal quotient of | C2×C3×ΓL(1,𝔽8)×C277 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17451, 10);