Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of ΓL(2,𝔽4) |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | C2×ΓL(2,𝔽4) |
| unambiguous semidirect product | (C2×C3)⋊S5 |
| unambiguous semidirect product | (C2×A5)⋊D3 |
| Fitting | C2×C3 ⋊ S5 |
| socle | C2×C3×A5 ⋊ C2 |
| derived | C3×A5 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(2,𝔽4) |
| commutator, cosocle, hypocenter | C3×A5 | C22 |
| Frattini | C1 | G |
| Fitting | C2×C3 | S5 |
| socle | C2×C3×A5 | C2 |
| LCS | G ▷ C3×A5 | |
| UCS | C1 ◁ C2 | |
129 conjugacy classes of subgroups.
| transitive degree | μt(G) = 30 |
24 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(2,𝔽4)⋊C4, #13088285, #13088287, … |
| maximal normal subgroup of | ΓL(2,𝔽4)⋊C4, #13088285, #13088287, … |
| maximal quotient of | #13088311, C4×ΓL(2,𝔽4), C4⋊ΓL(2,𝔽4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 770);