Finite Group DatabasebetaFinite Group Databasebeta| order | 1080 |
| isoclinism | isoclinic class of (C3⋊D9)⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C3⋊D9)⋊AGL(1,𝔽5) |
| unambiguous semidirect product | ΓL(1,𝔽16)⋊D9 |
| unambiguous semidirect product | (C9⋊AGL(1,𝔽5))⋊D3 |
| Fitting, derived, Hall | C3×C5×C9 ⋊ C2×C4 |
| Sylow | C3×C9 ⋊ C2×AGL(1,𝔽5) |
| Sylow | C5 ⋊ (C3⋊D9)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C3×C5×C9 | C2×C4 |
| Frattini | C3 | ΓL(1,𝔽16)⋊D3 |
| socle | C32×C5 | C4×D3 |
| prime radical | D5×C3⋊D9 | C2 |
| cosocle | C3×C9×D5 | C22 |
| LCS | G ▷ C3×C5×C9 | |
| UCS | C1 | |
116 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 90 |
Polycyclic presentation:
2^3,3^2,5,3|a^2=c,e^3=g,d^a=d^2,f^a=f^2,d^b=d^2,e^b=e^2g^2,g^b=g^2,f^c=f^451 conjugacy classes — view the full character table.
| maximal subgroup of | #72222410, #72222411, D3×D9×AGL(1,𝔽5), … |
| maximal normal subgroup of | #72222411, D3×D9×AGL(1,𝔽5) |
| maximal quotient of | #72222410, #72247383, #72254827 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1080, 277);