Finite Group DatabasebetaFinite Group Databasebeta| order | 1080 |
| isoclinism | isoclinic class of ΓL(1,𝔽16)⋊(C3⋊D3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | ΓL(1,𝔽16)⋊(C3⋊D3) |
| Fitting, socle, derived, Hall | C33×C5 ⋊ C2×C4 |
| Sylow | C33 ⋊ C2×AGL(1,𝔽5) |
| Sylow | C5 ⋊ #6289 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C33×C5 | C2×C4 |
| prime radical | D5×C3⋊(C3⋊D3) | C2 |
| cosocle | C33×D5 | C22 |
| LCS | G ▷ C33×C5 | |
| UCS | C1 | |
272 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 49 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:
2^3,3^3,5|a^2=c,d^a=d^2,e^a=e^2,g^a=g^2,d^b=d^2,e^b=e^2,f^b=f^2,g^c=g^451 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1080, 508);