Finite Group DatabasebetaFinite Group Databasebeta| order | 1134 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C3⋊(D7⋊H(3,𝔽3)) |
| unambiguous semidirect product | D21⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×C3⋊AGL(1,𝔽7))⋊C3 |
| Fitting | C33×C7 ⋊ C2×C3 |
| Hall | C3×C7⋊H(3,𝔽3) ⋊ C2 |
| Sylow | C7 ⋊ D3×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×C3⋊AGL(1,𝔽7) |
| commutator, socle, cosocle | C32×C7 | C2×C32 |
| Fitting | C33×C7 | C2×C3 |
| hypercenter | C32 | C3⋊AGL(1,𝔽7) |
| hypocenter | C3×C7 | C2×H(3,𝔽3) |
| LCS | G ▷ C32×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C3 ◁ C32 | |
122 conjugacy classes of subgroups.
| transitive degree | μt(G) = 126 |
Polycyclic presentation:
2,3^4,7|e^a=e^2,f^a=f^6,c^b=cd,f^b=f^460 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1134, 171);