Finite Group DatabasebetaFinite Group Databasebeta| order | 1458 |
| isoclinism | isoclinic class of C3⋊(C3⋊(C3⋊((C3⋊D3)⋊C3))) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C37 |
| unambiguous semidirect product | C3⋊(C3⋊(C3⋊((C3⋊D3)⋊C3))) |
| Fitting, Sylow | C33×H(3,𝔽3) ⋊ C2 |
| derived | C35 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C35 | C2×C3 |
| Frattini | C3 | C3×C3⋊(C3⋊(C3⋊D3)) |
| Fitting | C33×H(3,𝔽3) | C2 |
| socle | C34 | C3×D3 |
| LCS | G ▷ C35 | |
| UCS | C1 | |
5258 conjugacy classes of subgroups.
| transitive degree | μt(G) = 243 |
Polycyclic presentation:
2,3^6|b^a=b^2,c^a=bcf^2g,d^a=d^2,e^a=e^2,f^a=f^2,g^a=g^2,c^b=cg153 conjugacy classes — view the full character table.
| maximal subgroup of | #72221887, #72221889, C3⋊D3×C3⋊((C3⋊D3)⋊C3), … |
| maximal normal subgroup of | #72221887, #72221889, C3⋊D3×C3⋊((C3⋊D3)⋊C3), … |
| maximal quotient of | C2×C3⋊(C3⋊(C3⋊((C3⋊D3)⋊C3))) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 1758);