Finite Group DatabasebetaFinite Group Databasebeta| order | 1134 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊D27 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊D27 |
| Fitting, derived | C7×C27 ⋊ C2×C3 |
| Hall | ΓL(1,𝔽8)×C27 ⋊ C2 |
| Sylow | C7 ⋊ C3×D27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×C27 | C2×C3 |
| Frattini | C9 | C3⋊AGL(1,𝔽7) |
| socle | C3×C7 | C3×D9 |
| LCS | G ▷ C7×C27 | |
| UCS | C1 | |
38 conjugacy classes of subgroups.
| transitive degree | μt(G) = 189 |
Polycyclic presentation:
2,3^2,7,3^2|c^3=e^2,e^3=f^2,c^a=c^2ef,d^a=d^6,e^a=e^2f,f^a=f^2,d^b=d^272 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×D27, #72249083 |
| maximal normal subgroup of | AGL(1,𝔽7)×D27, #72249083 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1134, 43);