Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of ΓL(2,𝔽4) |
| automorphism | Aut(G) = D3×S5 (order 720) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| general semilinear | ΓL(2,𝔽4) |
| unambiguous semidirect product | C3⋊S5 |
| unambiguous semidirect product | A5⋊D3 |
| Fitting | C3 ⋊ S5 |
| socle, derived | C3×A5 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | C3×A5 | C2 |
| Fitting | C3 | S5 |
| LCS | G ▷ C3×A5 | |
| UCS | C1 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 15 |
A minimal faithful permutation representation — 2 generators on 8 points:
(1,2)(4,5,7)(6,8);(1,2,3)(4,6,5)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 2 | 15 | 20 | 24 | 24 | 24 | 30 | 30 | 40 | 60 | 90 |
| order | 1 | 3 | 2 | 3 | 5 | 15 | 15 | 2 | 6 | 3 | 6 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 |
| χ3 | 2 | -1 | 2 | 2 | 2 | -1 | -1 | 0 | -1 | -1 | 0 | 0 |
| χ4 | 4 | 4 | 0 | 1 | -1 | -1 | -1 | 2 | 0 | 1 | -1 | 0 |
| χ5 | 4 | 4 | 0 | 1 | -1 | -1 | -1 | -2 | 0 | 1 | 1 | 0 |
| χ6 | 5 | 5 | 1 | -1 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 |
| χ7 | 5 | 5 | 1 | -1 | 0 | 0 | 0 | -1 | 1 | -1 | -1 | 1 |
| χ8 | 6 | 6 | -2 | 0 | 1 | 1 | 1 | 0 | -2 | 0 | 0 | 0 |
| χ9 | 6 | -3 | -2 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | ||
| χ10 | 6 | -3 | -2 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | ||
| χ11 | 8 | -4 | 0 | 2 | -2 | 1 | 1 | 0 | 0 | -1 | 0 | 0 |
| χ12 | 10 | -5 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
| maximal subgroup of | D3×S5, C2×ΓL(2,𝔽4), C9⋊S5, … |
| maximal normal subgroup of | D3×S5, C2×ΓL(2,𝔽4), C3×ΓL(2,𝔽4), … |
| maximal quotient of | C3⋊(A5⋊C4), C3⋊#6728, C3⋊#6729, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 120);