Finite Group DatabasebetaFinite Group Databasebeta| order | 1134 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C36 |
| direct product | C2×C33×ΓL(1,𝔽8) |
| unambiguous semidirect product | (C2×C7)⋊C34 |
| unambiguous semidirect product | (C2×C3×C7)⋊C33 |
| Fitting, socle | C2×C33×C7 ⋊ C3 |
| derived, Sylow | C7 ⋊ C2×C34 |
| Hall | C2×C7 ⋊ C34 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C33 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C2×C34 |
| Frattini | C1 | G |
| Fitting, socle | C2×C33×C7 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C33 | |
848 conjugacy classes of subgroups.
| transitive degree | μt(G) = 378 |
270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C32×ΓL(1,𝔽8)×AGL(1,𝔽7) |
| maximal quotient of | C32×ΓL(1,𝔽8)×AGL(1,𝔽7) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1134, 244);