Finite Group DatabasebetaFinite Group Databasebeta| order | 1458 |
| isoclinism | isoclinic class of #13091127 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C32 |
No aliases for this group.
| Fitting, Sylow | #10624752 ⋊ C2 |
| derived | C3×AΣL(1,𝔽27) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×AΣL(1,𝔽27) | C2×C3 |
| Frattini | C33 | C3×C3⋊D3 |
| Fitting | #10624752 | C2 |
| socle | C3 | #92300 |
| LCS | G ▷ C3×AΣL(1,𝔽27) | |
| UCS | C1 | |
242 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
Polycyclic presentation:
2,3^6|b^3=g,c^3=g,d^3=g,b^a=b^2c^2de^2fg^2,c^a=d^2g^2,d^a=c^2g^2,e^a=fg,f^a=eg,g^a=g^2,c^b=ce,d^b=df,e^b=eg,f^c=fg,e^d=eg42 conjugacy classes — view the full character table.
| maximal subgroup of | #72243664, #72243665, C3×#13091127, … |
| maximal normal subgroup of | C3×#13091127, #72243770 |
| maximal quotient of | C3×#13091127 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 1201);