Finite Group DatabasebetaFinite Group Databasebeta| order | 1134 |
| isoclinism | isoclinic class of C3⋊((C3⋊D3)⋊ΓL(1,𝔽8)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C3⋊((C3⋊D3)⋊ΓL(1,𝔽8)) |
| Fitting, derived | C33×C7 ⋊ C2×C3 |
| Hall | C3×C7⋊H(3,𝔽3) ⋊ C2 |
| Sylow | C7 ⋊ C3⋊((C3⋊D3)⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C33×C7 | C2×C3 |
| Frattini | C3 | C3⋊D3×ΓL(1,𝔽8) |
| socle | C32×C7 | C3×D3 |
| LCS | G ▷ C33×C7 | |
| UCS | C1 | |
134 conjugacy classes of subgroups.
| transitive degree | μt(G) = 189 |
Polycyclic presentation:
2,3^4,7|c^a=c^2,d^a=d^2,e^a=e^2,d^b=de,f^b=f^451 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×C3⋊((C3⋊D3)⋊C3), #72269696 |
| maximal normal subgroup of | ΓL(1,𝔽8)×C3⋊((C3⋊D3)⋊C3), #72269696 |
| maximal quotient of | #72249177 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1134, 179);