Finite Group DatabasebetaFinite Group Databasebeta| order | 1080 |
| isoclinism | isoclinic class of ΓL(2,𝔽4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×ΓL(2,𝔽4) |
| unambiguous semidirect product | C3⋊(C3×S5) |
| unambiguous semidirect product | A5⋊(C3×D3) |
| Fitting | C32 ⋊ S5 |
| socle | C32×A5 ⋊ C2 |
| derived | C3×A5 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ΓL(2,𝔽4) |
| commutator, cosocle, hypocenter | C3×A5 | C2×C3 |
| Frattini | C1 | G |
| Fitting | C32 | S5 |
| socle | C32×A5 | C2 |
| LCS | G ▷ C3×A5 | |
| UCS | C1 ◁ C3 | |
109 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 109 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 30 |
36 conjugacy classes — view the full character table.
| maximal subgroup of | C3×AΓL(2,𝔽4), D3×ΓL(2,𝔽4), C3×A4⋊S5, … |
| maximal normal subgroup of | D3×ΓL(2,𝔽4), C3×D3×S5, #72254793 |
| maximal quotient of | C3×AΓL(2,𝔽4), C3×A4⋊S5 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1080, 489);