Finite Group DatabasebetaFinite Group Databasebeta| order | 1134 |
| isoclinism | isoclinic class of ΓL(1,𝔽64) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C3×ΓL(1,𝔽64) |
| Fitting | C3×C7×C9 ⋊ C2×C3 |
| derived | C7×C9 ⋊ C2×C32 |
| Hall | C3×#4428 ⋊ C2 |
| Sylow | C7 ⋊ C3×AGL(1,ℤ/9) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ΓL(1,𝔽64) |
| commutator, cosocle, hypocenter | C7×C9 | C2×C32 |
| Frattini | C3 | C3×D3×ΓL(1,𝔽8) |
| Fitting | C3×C7×C9 | C2×C3 |
| socle | C32×C7 | C3×D3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 ◁ C3 | |
92 conjugacy classes of subgroups.
| transitive degree | μt(G) = 126 |
66 conjugacy classes — view the full character table.
| maximal subgroup of | C3×ΓL(1,𝔽8)×AGL(1,ℤ/9), #72229912, #72229913, … |
| maximal normal subgroup of | C3×ΓL(1,𝔽8)×AGL(1,ℤ/9), #72229912, #72229913, … |
| maximal quotient of | #72229913 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1134, 150);