Finite Group DatabasebetaFinite Group Databasebeta| order | 1134 |
| isoclinism | isoclinic class of C3⋊((C3⋊D3)⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C32 |
| direct product | C7×C3⋊((C3⋊D3)⋊C3) |
| unambiguous semidirect product | C3⋊(C7×(C3⋊D3)⋊C3) |
| Fitting, Hall | C3×C7×H(3,𝔽3) ⋊ C2 |
| derived | C33 ⋊ C2×C3×C7 |
| Sylow | C3×H(3,𝔽3) ⋊ C2×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | C3⋊((C3⋊D3)⋊C3) |
| commutator, cosocle, hypocenter | C33 | C2×C3×C7 |
| Frattini | C3 | C3×C7×C3⋊D3 |
| Fitting | C3×C7×H(3,𝔽3) | C2 |
| socle | C32×C7 | C3×D3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C7 | |
134 conjugacy classes of subgroups.
| transitive degree | μt(G) = 189 |
147 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×C3⋊((C3⋊D3)⋊C3), D7×C3⋊((C3⋊D3)⋊C3) |
| maximal normal subgroup of | ΓL(1,𝔽8)×C3⋊((C3⋊D3)⋊C3), D7×C3⋊((C3⋊D3)⋊C3) |
| maximal quotient of | C7×#92300 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1134, 217);